Stellar movements and the structure of the universe

t

MACMII.LAX AND CO., LIMITED

LONDON . ROMHAY . C.MXIII A
MELBOURNE

THK MA< "Ml LI. AN COMPANY

Ni:\V YORK . BOSTON . CHICAGO

DALLAS . <AN KKANCISCO

THE MACMILLAN CO. OF CANADA, LTD.

TOR'

Plate I,

GR,KAT STAR-CLOUD IN SAGITTARIUS.

STELLAR MOVEMENTS

AND THE STRUCTURE

OF THE UNIVERSE

BY

A. S. EDDINGTON

'\

M.A. (CANTAB.), M.Sc. (MANCHESTER), B.Sc. (Lo.\n.), F.R.S.

Plnmian Professor of Astronomy, University of Cambridge

MACMILLAN AND CO., LIMITED
ST. MARTIN'S STREET, LONDON

1914

COPYR1GH7*

PREFACE

THE purpose of this monograph is to give an account of
the present state of our knowledge of the structure of the
sidereal universe. This branch of astronomy has become
especially prominent during the last ten years ; and many
new facts have recently been brought to light. There is
every reason to hope^that the next few years will be equally
fruitful ; and it may seem hazardous at the present stage to
attempt a general discussion of our knowledge. Yet perhaps
at a time like the present, when investigations are being
actively prosecuted, a survey of the advance made may be
especially helpful.

The knowledge that progress will inevitably lead to a
readjustment of ideas must instil a writer with caution;
but I believe that excessive caution is not to be desired.
There can be no harm in building hypotheses, and weaving
explanations which seem best fitted to our present partial
knowledge. These are not idle speculations if they help
us, even temporarily, to grasp the relations of scattered
fjicts, and to organise our knowledge.

No attempt has been made to treat the subject histori-
cally. I have preferred to describe the results of inves-

300340

vi PREFACE

tigations founded on the most recent data rather than early
pioneer researches. One inevitable result I particularly
regret ; many of the workers who have prepared the way
for recent progress receive but scanty mention. Sir W.
Herschel, Kobold, Seeliger, Newcomb and others would
have figured far more prominently in a historical account.
But it was outside my purpose to describe the steps by
which knowledge has advanced ; it is the present situation
that is here surveyed.

So far as practicable I have endeavoured to write for
the general scientific reader. It was impossible, without
too great a sacrifice, to avoid mathematical arguments
altogether ; but the greater part of the mathematical
analysis has been segregated into two chapters (VII and
X). Its occasional intrusion into the remaining chapters
will, it is hoped, not interfere with the readable character
of the book.

I am indebted to Prof. G. E. Hale for permission to re-
produce the two photographs of nebulae (Plate 4), taken by
Mr. G. W. Ritchey at the Mount Wilson Observatory, and
to the Astronomer Royal, Dr. F. W. Dyson, for the three re-
maining plates taken from the Franklin-Adams Chart of the
Sky. This represents but a small part of my obligation to
I )r. Dyson ; at one time and another nearly all the subjects
treated in this book have been discussed between us, and I
make no attempt to discriminate the idcns wliir-h I owe
to him. There are many other astronomers from whose
conversation, consciously or unconsciously, I have drawn
material for this work.

PREFACE vii

I have to thank Mr. , P. J. Melotte of the Royal
Observatory, Greenwich, who kindly prepared the three
Franklin- Adams photographs for reproduction.

Dr. S. Chapman, Chief Assistant at the Royal Observa-
tory, Greenwich, has kindly read the proof-sheets, and I am
grateful to him for his careful scrutiny and advice.

I also desire to record my great indebtedness to the
Editor of this series of monographs, Prof. R. A. Gregory,
for many valuable suggestions and for his assistance in
passing this work through the press.

A. S. EDDINGTON.

THE OBSERVATORY, CAMBRIDGE.
April, 1914.

CONTENTS

CHAPTER I

PAOF.

THE DATA OF OBSERVATION 1

CHAPTER II

GENERAL OUTLINE . 30

CHAPTER III

THE NEAREST STARS 40

CHAPTER IV

-MOVING CLUSTERS 54

CHAPTER V

THE SOLAR MOTION 71

CHAPTER VI

THE TWO STAR STREAM- 86

CHAPTER VII

THE TWO STAR STREAMS MATHEMATICAL THEORY 127

CHAPTER VIII

PHENOMENA ASSOCIATED WITH SPECTRAL TYPE 154

CHAPTER IX

COUNTS OF STARS 184

x CONTENTS

CHAPTER X

GENERAL STATISTICAL INVESTIGATIONS 201

CHAPTER XI

THE MILKY WAY, STAR-CLUSTERS AND NEBULAE 232

CHAPTER XII

DYNAMICS OF THE STELLAR SYSTEM 246

INDEX 263

LIST OF ILLUSTRATIONS

PLATES

/

PLATE I. The Great Star-cloud in Sagittarius Frontispiece

,, II. Nebulous dark space in Corona Austrina facing paye 236
,, III. The Greater Magellanic Cloud ,, 240

(Nebula, Canes Venatici. N.G.C. 5194-5 \ 249

' iNebula, Coma Berenices. H.V. 24 /

FIGURES

PAGI

1. Hypothetical Section of the Stellar System 31

2. Moving Cluster in Taurus. (Boss.) 57

3. Geometrical Diagram 59

4. Moving Cluster of " Orion "Stars in Perseus 64

5. Simple Drift Curves 88

6. Observed Distribution of Proper Motions. (Groombridge Cata-

logue R. A. 14*> to 18 h , Dec. +38 to +70 : .) .... 89

7. Calculated Distributions of Proper Motions 92

8. Observed and Calculated Distributions of Proper Motions.

(Boss, Region VIII.) 93

9. Diagrams for the Proper Motions of Boss's " Preliminary

General Catalogue " 95, 96, 97, 98

10. Convergence of the Directions of Drift I. from the 17 Regions . 99

11. Convergence of the Directions of Drift II. from the 17 Regions . 99

12. Geometrical Diagram 101

13. Distribution of Large Proper Motions (Dyson) .... 105

14. Mean Proper Motion Curves 112

xi

xii LIST OF ILLUSTRATIONS

Hi;. PAGE

15. Mean Proper Motion Curve (Region of Boss's Catalogue) . . 116

16. Comparison of Two-drift and Three-drift Theories . . .122

17. Distribution of Drifts along the Equator (Halm) .... 123

18. Comparison of Two-drift and Ellipsoidal Hypotheses . . . 135

19. Absolute Magnitudes of Stars (Russell) 171

20. Number of Stars brighter than each Magnitude for Eight Zones

(Chapman and Melotte) 188

21. Curves of Equal Frequency of Velocity 220

22. Geometrical Diagram 249

STELLAR MOVEMENTS AND THE
STRUCTURE OF THE UNIVERSE

CHAPTER I

THE DATA OF OBSERVATION

IT is estimated that the number of stars which could
be revealed by the most powerful telescopes now in use
amounts to some hundreds of millions. One of the princi-
pal aims of stellar astronomy is to ascertain the relations
and associations which exist among this multitude of
individuals, and to study the nature and organisation of
the great system which they constitute. This study is as
yet in its infancy ; and, when we consider the magnitude
of the problem, we shall scarcely expect that progress
towards a full understanding of the nature of the sidereal
universe will be rapid. But active research in this branch
of astronomy, especially during the last ten years, has led
to many results, which appear to be well established. It
has become possible to form an idea of the general dis-
tribution of the stars through space, and the general
character of their motions. Though gaps remain in our
knowledge, and some of the most vital questions are as
yet without an answer, investigation along many different
lines has elicited striking facts that may well be set down.
It is our task in the following pages to coordinate these
results, and review the advance that has been made.

B

MOVEMENTS CHAP.

Until recent years the study of the bodies of the solar
system formed by far the largest division of astronomy ;
but with that branch of the subject we have nothing to do
here. From our point of view the whole solar system is
only a unit among myriads of similar units. The system
of the stars is on a scale a million-fold greater than that of
the planets ; and stellar distances exceed a million times
the distances of the comparatively well-known bodies
which circulate round the Sun. Further, although we
have taken the stars for our subject, not all branches of
stellar astronomy fall within the scope of this book. It
is to the relations between the stars to the stars as a
society that attention is here directed. We are not con-
cerned with the individual peculiarities of stars, except in
so far as they assist in a broad classification according to
brightness, stage of development, and other properties.
Accordingly, it is not proposed to enter here into the
more detailed study of the physical characters of stars ;
and the many interesting phenomena of Variable Stars
and Novae, of binary systems, of stellar chemistry and
temperatures are foreign to our present aim.

The principal astronomical observations, on which the
whole superstructure of fact or hypothesis must be based,
may be briefly enumerated. The data about a star which
are useful for our investigations are :

1. Apparent position in the sky.

2. Magnitude.

3. Type of spectrum, or Colour.

4. Parallax.

5. Proper motion.

6. Radial velocity.

In addition, it is possible in some rather rare cases to find
the mass or density of a star. This is a matter of import-
ance, for presumably one star can only influence another
by means of its gravitational attraction, which depends
on the masses.

1 THE DATA OF OBSERVATION 3

This nearly exhausts the list of characteristics which
are useful in investigating the general problems of stellar
distribution.* It is only in the rarest cases that the com-
plete knowledge of a star, indicated under the foregoing
six heads, is obtainable ; and the indirect nature of most of
the processes of investigation that are adopted is due to
the necessity of making as much use as possible of the very
partial knowledge that we do possess.

The observations enumerated already will now be con-
sidered in order. The apparent position in the sky needs
no comment ; it can always be stated with all the accuracy
requisite.

Magnitude. The magnitude of a star is a measure of
its apparent brightness ; unless the distance is known,
we are not in a position to calculate the intrinsic or
absolute brightness. Magnitudes of stars are measured
on a logarithmic scale. Starting from a sixth magnitude
star, which represents an arbitrary standard of brightness
of traditional origin, but now fixed with sufficient pre-
cision, a star of magnitude 5 is one from which we receive

2 '5 12 times more light. Similarly, each step of one magni-
tude downwards or upwards represents an increase or
decrease of light in the ratio 1 : 2*5 12. f The number is so
chosen that a difference of five magnitudes corresponds to
a light ratio of 100 = (2*5 12) 5 . The general formula is

= - --

where

.Lj, L 2 are the intensities of the light from two stars
mj, m 2 are their magnitudes.

Magnitude classifications are of two kinds : photometric
(or visual), and photographic ; for it is often found that of

* We should perhaps add that the separations and periods of binary stars
are also likely to prove useful data.

f It is necessary to warn the reader that there are magnitude systems still
in common use that of the Bonn Durchmusterung, that used by many
double-star observers, and even the magnitudes of Boss's Preliminary
General Catalogue (1910) which do not conform to this scale.

B 2

4 STELLAR MOVEMENTS CHAP.

two stars the one which appears the brighter to the eye
leaves a fainter image on a photographic plate. Neither of
the two systems has been defined very rigorously; for, when
stars are of different colours, a certain amount of personality
exists in judging equality of light by the eye, and, if a
photographic plate is used, differences may arise depending
on the colour-sensitiveness of the particular kind of plate,
or on the chromatic correction of the telescope object-
glass. As the accuracy of magnitude-determinations im-
proves, it will probably become necessary to adopt more
precise definitions of the visual and photographic scales ;
but at present there appears to be no serious want of
uniformity from this cause. But the distinction between
the photometric and photographic magnitudes is very
important, and the differences are large. The bluer the
colour of a star, the greater is its relative effect on the
photographic plate. A blue star and a red star of the
same visual brightness may differ photographically by as
much as two magnitudes.

The use of a logarithmic scale for measuring brightness
possesses many advantages ; but it is liable to give a
misleading impression of the real significance of the
numbers thus employed. It is not always realised how
very rough are the usual measures of stellar brightness.
If the magnitude of an individual star is not more than
IU T in error, we are generally well satisfied ; yet this
means an error of nearly 10 per cent, in the light-intensity.
Interpreted in that way it seems a rather poor result.
An important part of stellar investigation is concerned
with counts of the number of stars within certain limits of
magnitude. As the number of stars increases about
three-fold for each successive step of one magnitude, it is
clear that all such work will depend very vitally on the
absence of systematic error in the adopted scale of
magnitudes; an error of two- or three- tenths of a magni-
tude would affect the figures profoundly. The establish-

i THE DATA OF OBSERVATION 5

ment of an accurate magnitude-system, with sequences of
standard stars, has been a matter of great difficulty, and
it is not certain that even now a sufficiently definitive
system has been reached. The stars which coine under
notice range over more than twenty magnitudes, corre-
sponding to a light ratio of 100,000,000 to 1. To
sub-divide such a range without serious cumulative error
would be a task of great difficulty in any kind of physical
measurement.

The extensive researches of the Harvard Observatory,
covering both hemispheres of the sky, are the main basis
of modern standard magnitudes. The Harvard sequence
of standard stars in the^neighbourhood of the North Pole,
extending by convenient steps as far as magnitude 21,
at the limit reached with the 60-inch reflector of the
Mount Wilson Observatory, now provides a suitable scale
from which differential measures can be made. The
absolute scale of the Harvard sequence of photographic
magnitudes has been independently tested by F. H. Scares,
at Mount Wilson, and S. Chapman and P. J. Melotte, at
Greenwich. Both agree that from the tenth to the
fifteenth magnitudes the scale is sensibly correct. But
according to Scares a correction is needed between the
second and ninth magnitudes, l m '00 on the Harvard scale
being equivalent to l m *07 absolute. If this result is
correct, the error introduced into statistical discussions
must be quite appreciable.

For statistical purposes there are now available deter-
minations of the magnitudes of the stars in bulk made at
Harvard, Potsdam, Gottingen, Greenwich and Yerkes
Observatories. The revised Harvard photometry gives the
visual magnitudes of all stars down to about 6 m '5 ; the
Potsdam magnitudes, also visual, carry us in a more limited
part of the sky as far as magnitude 7 m '5. The Gottingen
determinations, which are absolute determinations, inde-
pendent of but agreeing very fairly with the Harvard

6 STELLAR MOVEMENTS CHAP.

sequences, provide photographic magnitudes over a large
zone of the sky for stars brighter than 7 m< 5. The Yerkes
investigation gives visual and photographic magnitudes
of the stars within 17 of the North Pole down to 7 m> 5.
A series of investigations at Greenwich, based on the
Harvard sequences, provides statistics for the fainter stars
extending as far as magnitude 17, and is a specially
valuable source for the study of the remoter parts of the
stellar system.

So far we have been considering the apparent brightness
of stars and not their intrinsic brightness. The latter
quantity can be calculated when the distance of the star
is known. We shall measure the absolute luminosity in
terms of the Sun as unit. The brightness of the Sun has
been measured in stellar magnitudes and may be taken to
be 26 m 'l, that is to say it is 26*1 magnitudes brighter
than a star of zero magnitude.* From this the luminosity
L of a star, the magnitude of which is m, and parallax
is &", is given by

Iog 10 L = 0'2 - 0'4 x w -2 log lo Z.

The absolute magnitudes of stars differ nearly as widely
as their apparent magnitudes. The feeblest star known is
the companion to Groombridge 34, which is eight magni-
tudes fainter than the Sun. Estimates of the luminosities
of the brightest stars are usually very uncertain ; but, to
take only results which have been definitely ascertained,
the Cepheid Variables are on an average seven magnitudes
brighter than the Sun. Probably this luminosity is
exceeded by many of the Orion type stars. There is thus
a range of at least fifteen magnitudes in the intrinsic
brightness, or a light ratio of 1,000,000 to 1.

* The most recent researches give a value -26*5 (Ceraski, Annul* of the
Obwvatory of Moscow, 1911). It is best, however, to regard the unit of
luminosity as a conventional unit, roughly representing the Sun. ami defined
by the formula, rather than to keep changing tin- measures of stellar
luminosity every time a better determination of the Sun's stellar magnitude
is mad*-.

i THE DATA OF OBSERVATION 7

Type of Spectrum. For the type of spectrum
various systems of classification have been used by astro-
physicists, but that of the Draper Catalogue of Harvard
Observatory is the most extensively employed in work on
stellar distribution. This is largely due to the very
complete classification of the brighter stars that has been
made on this system. The classes in the supposed order
of evolution are denoted by the letters :

O, B, A, F, G, K, M, N.

A continuous gradation is recognised from to M, and a
more minute sub-division is obtained by supposing the
transition from one class to the next to be divided into
tenths. Thus B5A, usually abbreviated to B5, indicates a
type midway between B and A ; G2 denotes a type
between Gr and K, but more closely allied to the former.
It is usual to class as Type A all stars from AO to A9 ; but
presumably it would be preferable to group together the
stars from B6 to A5, and this principle has occasionally
been adopted.

For our purposes it is not generally necessary to consider
what physical peculiarities in the stars are represented by
these letters, as the knowledge is not necessary for discussing
the relations of motion and distribution. All that we require
is a means of dividing the stars according to the stage of
evolution they have attained, and of grouping the stars
with certain common characteristics. It may, however, be
of interest to describe briefly the principles which govern
the classification, and to indicate the leading types of
spectrum.

Tracing an imaginary star, as it passes through the
successive stages of evolution from the earliest to the
latest, the changes in its spectrum are supposed to pursue
the following course. At first the spectrum consists
wholly of diffuse bright bands on a faint continuous back-
ground. The bands become fewer and narrower, and faint

8 STELLAR MOVEMENTS CHAP.

absorption lines make their appearance ; the first lines seen
are those of the various helium series, the well-
known Balmer hydrogen series, and the " additional "
or "sharp" hydrogen series.* The last is a spectrum
which had been recognised in the stars by E. C. Pickering
in 1896 but was first produced artificially by A. Fowler
in 1913. The bright bands now disappear, and in the
remaining stages the spectrum consists wholly of absorp-
tion lines and bands, except in abnormal individual
stars, which exceptionally show bright lines. The next
phase is an enormous increase in the intensity of the
true hydrogen spectrum, the lines becoming very wide
and diffused ; the other lines disappear. The lines H
and K of calcium and other solar lines next become
evident and gain in intensity. After this the hydrogen
lines decline, though long remaining the leading feature of
the spectrum ; first a stage is reached in which the calcium
spectrum becomes very intense, and afterwards all the
multitudinous liues of the solar spectrum are seen. After
passing the stage reached by our Sun, the chief feature is
a shortening of the spectrum from the ultraviolet end, a
further fading of the hydrogen lines, an increase in the
number of fine absorption lines, and finally the appearance
of bands due to metallic compounds, particularly the flutings
of titanium oxide. The whole spectrum ultimately approxi-
mates towards that of sunspots.

Guided by these principles, we distinguish eight leading
types, between which, however, there is a continuous series
of gradations.

TYPE (WOLF-RAYET TYPE) The spectrum consists of
bright bands on a faint continuous background ; of these
the most conspicuous have their centres at \\ 4633,
4651, 4686, 5693, and 5813. The type is divided into

* The practical worU of A. Fowler and the theoretical researches of N. I5 tin-
leave little doubt that this spectrum is due to helium, notwithstanding its
simple numerical relation to the hydrogen spectrum.

i THE DATA OF OBSERVATION 9

five divisions, Oa, Ob, Oe, Od, and Oe, marked by varying
intensities and widths of the bands. Further, in Od and
Oe dark lines, chiefly belonging to the helium and helium-
hydrogen series, make their appearance.

TYPE B (ORION TYPE) This is often called the helium
type owing to the prominence of the lines of that
element. In addition there are some characteristic lines
the origin of which is unknown, as well as both the " sharp "
and Balmer series. The bright bands seen in Type are no
longer present; in fact, they disappear as early as the
sub-division OeoB, which is therefore usually reckoned the
starting point of the Orion type.

TYPE A (SiRiAN TYPE). The Balmer series of hydro-
gen is present in great intensity, and is far the most
conspicuous feature of the spectrum. Other lines are
present, but they are relatively faint.

TYPE F (CALCIUM TYPE). The hydrogen series is still
very conspicuous, but not so strong as in the preceding
type. The narrow H and K lines of calcium have become
very prominent, and characterise this spectrum.

TYPE G (SOLAR TYPE). The Sun may be regarded as
a typical star of this class, the numerous metallic lines
having made their appearance.

TYPE K. The spectrum is somewhat similar to the
last. It is mainly distinguished by the fact that the
hydrogen lines, which are still fairly strong in the G stars,
are now fainter than some of the metallic lines.

TYPE M. The spectrum is now marked by the appear-
ance of flutings, due to titanium oxide. It is remarkable
that the spectrum should be dominated so completely by
this one substance. Two successive stages are recognised,
indicated by the sub-divisions Ma and Mb. The long-
period variable stars, which show bright hydrogen lines in
addition to the ordinary characteristics of Type M, form
the class Md.

TYPE N. The regular progression of the types

io STELLAR MOVEMENTS CHAP.

terminates with Mb. There is no transition to Type N,
and the relation of this to the foregoing types is uncertain.
It is marked by characteristic flu tings attributed to com-
pounds of carbon. The stars of both the M and N types
are of a strongly reddish tinge.

It is sometimes convenient to use the rather less detailed
classification of A. Secchi. Strictly speaking his system
relates to the visual spectrum, and the Draper notation to
the photographic ; but the two can well be harmonised.

Secchi's Type I. includes Draper B and A

II. F, G, andK

III. M

TV N

>j -"-' -^

As comparatively few of the stars in any catalogue belong
to the last two types, this classification is practically a
separation into two groups, which are of about equal size.
This is a very useful division when the material is too
scanty to admit of a more extended discussion.

From time to time there are indications that the Draper
classification has not succeeded in separating the stars into
really homogeneous groups. According to Sir Norman
Lockyer, there are stars of ascending temperature and of
descending temperature in practically every group ; so
that, for example, the stars enumerated under K are a
mixture of two classes, one in a very early, the other in a late,
stage of evolution. In the case of Type B, H. Ludendorff 1
has found considerable systematic differences in the
measured radial motions of the stars classed by Lockyer as
ascending and descending respectively (pointing, however,
to real differences not of motion but of physical state,
which have introduced an error into the spectroscopic
measurements). E. Hertzsprung 2 has pointed out that
the presence or absence of the c character on Miss Maury's
classification (i.e., sharply defined absorption-lines) corre-
sponds to an important difference in the intrinsic
luminosities of the stars. Hitherto, however, it has

THE DATA OF OBSERVATION

ii

been usual to accept the Draper classification as at least
the most complete available for our investigations.

Colour-Index. Stars may be classified according to
colour as an alternative to spectral type. Both methods
involve dividing the stars according to the nature of
the light emitted by them, and thus have something in
common. Perhaps we might not expect a very close
correspondence between the two classifications ; for, whilst
colour depends mainly on the continuous background of
the spectrum, the spectral type is determined by the fine
lines and bands, which can have little direct effect on the
colour. Nevertheless a close correlation is found between
the two characters, owing no doubt to the fact that both
are intimately connected with the effective temperature of
the star.

The most convenient measure of colour is afforded by
the difference, photographic minus visual magnitude ; this
is called the colour-index. The relation between the
spectral type and the colour-index is shown below.

Colour index according to

Spectral Type.

King.

Schwarzschild.

m

m

Bo

-0-31

-0-64

Ao

o-oo

-0-32

Fo

+ 0-32

o-oo

Go

+ 0-71

+ 0-32

Ko

+ 1-17

+ 0-95

M

+ 1-68

+ 1-89

I

King's results 3 refer to the Harvard visual and photographic
scales ; Schwarzschild's 4 to the Gottingen photographic and
Potsdam visual determinations. Allowing for the constant
difference, depending on the particular type of spectrum
for which the photographic and visual magnitudes are

12 STELLAR MOVEMENTS CHAP.

made to agree, the two investigations confirm one another
closely.

The foregoing results are derived from the means of a
considerable number of stars, but the Table may be
applied to individual stars with considerable accuracy.
Thus the spectral type can be found when the colour-
index is known, and conversely. At least in the case of
the early type stars, the spectral type fixes the colour-
index with an average uncertainty of not more than O m 'l ;
for Types G and K larger deviations are found, but the
correlation is still a very close one.

Yet another method of classifying stars according to the
character of the light emitted is afforded by measures of
the "effective wave-length.'* If a coarse grating, consist-
ing of parallel strips or wires equally spaced, is placed in
front of the object-glass of a telescope, diffraction images
appear on either side of the principal image. These
diffraction images are strictly spectra, and the spot which
a measurer would select as the centre of the image will
depend on the distribution of intensity in the spectrum.
Each star will thus have a certain effective wave-length
which will be an index of its colour, or rather of the
appreciation of its colour by the photographic plate. For
the same telescope and grating the interval between the two
first diffracted images is a constant multiple of the effective
wave-length. The method was first used by K. Schwarz-
schild in 1895 ; and an important application of it was
made by Prosper Henry to determine the effect of
atmospheric dispersion on the places of the planet
Eros. It has been applied to stellar classification by
E. Hertzsprung. 5

Parallax. The annual motion of the earth around
the Sun causes a minute change in the direction in which
a star is seen, so that the star appears to describe a
small ellipse in the sky. This periodic displacement is
superposed on the uniform proper- motion of the star,

i THE DATA OF OBSERVATION 13

which is generally much greater in amount ; there is,
however, no difficulty in disentangling the two kinds of
displacement. Since we are only concerned with the
direction of the line joining the two bodies, the effect of
the earth's motion is the same as if the earth remained at
rest, and the star described an orbit in space equal to that
of the earth, but with the displacement reversed, so that
the star in its imaginary orbit is six months ahead of the
earth. This orbit is nearly circular ; but, as it is generally
viewed at an angle, it appears as an ellipse in the sky.
In any case the major axis of the ellipse is equal to the
diameter of the earth's orbit ; and, since the latter length
is known, a determination of its apparent or angular
magnitude affords a means of calculating the star's
distance. The parallax is defined as the angle subtended
by one astronomical unit (the radius of the earth's orbit)
at the distance of the star, and is equivalent to the major
semiaxis of the ellipse which the star appears to describe.

The measurement of this small ellipse is always made
relatively to some surrounding stars, for it is hopeless to
make absolute determinations of direction with the
necessary accuracy. The relative parallax which is thus
obtained needs to be corrected by the amount of the
average parallax of the comparison stars in order to obtain
the absolute parallax. This correction can only be
guessed from our general knowledge of the distances of
stars similar in magnitude and proper motion to the
comparison stars ; but as it could seldom be more than
0"*01, not much uncertainty is introduced into the final
result from this cause.

The parallax is the most difficult to determine of all the
quantities which we wish to know, and for only a very
few of the stars has it been measured with any approach
to certainty. Until some great advance is made in means
of measurement, all but a few hundreds of the nearest
stars must be out of range of the method. But so

i 4 STELLAR MOVEMENTS CHAP.

Laborious are the observations required, that even these
will occupy investigators for a long while. In general, the
published lists of parallaxes contain many that are ex-
tremely uncertain, and some that are altogether spurious.
Statistical investigations based on these are liable to be
very misleading. Nevertheless, it is believed that by
rejecting unsparingly all determinations but those of the
highest refinement, some important information can be
obtained, and in Chapter III. these results are discussed.
In addition, determinations which are not individually of
high accuracy may be used for finding the mean parallaxes
of stars of different orders of magnitude and proper motion,
provided they are sensibly free from systematic error ;
these at least serve to check the results found by less
direct methods.

The measurements are generally made either by photo-
graphy or visually with a heliometer. The former method
now appears to give the best results owing to the greater
focal-length of the instruments available. It has also the
advantage of using a greater number of comparison stars,
so that there is less chance of the correction to reduce to
absolute parallax being inaccurate. Some early determin-
ations with the heliometer are, however, still unsurpassed.
The meridian-circle is also used for this work, and consider-
able improvement is shown in the more recent results of
this method ; but we still think that meridian parallaxes
are to be regarded with suspicion, and have deemed it
best not to use them at all in Chapter III.

A convenient unit for measuring stellar distances is the
parsec* or distance which corresponds to a parallax of one

* The parsec, a p<>i tin.-uitrau-namr sujj^i-strd by H. H. Turner, will be
used throughout this book. Several different units of stellar distance have,
however, been employed by investigators. Kobold's Sternweite is identical
with the parsec. Seeliger's Siriusu'eite corresponds to a parallax 0"'2, and
( 'h;irli-r's Xiriometer to a million astronomical units or purallax 0"'206. The
light-year which, notwithstanding its inconvenience and irrelevance, has
sometimes crept from popular use into technical investigations, is e<]ual to
0'31 parsecs.

i THE DATA OF OBSERVATION 15

second of arc. This is equal to 206,000 astronomical units
or about 19,000,000,000,000 miles. The nearest fixed
star, a Centauri, is at a distance of 1*3 parsecs. There
are perhaps thirty or forty stars within a distance of five
parsecs, and, of course, the number at greater distances will
increase as the cube of the limiting distance, so long as
the distribution is uniform. But these nearest stars are
not by any means the brightest visible to us ; the range
in intrinsic luminosity is so great that the apparent magni-
tude is very little clue to the distance. A sphere of radius
thirty parsecs would probably contain 6000 stars ; but
the 6000 stars visible to the average eye are spread through
a far larger volume of space. It appears indeed that some
of the naked -eye stars are situated in the remotest parts
of the stellar system.

A parallax-determination may be considered first-class
if its probable error is as low as 0"'01. If, for instance,
the measured parallax is 0"'050"*01, it is an even chance
that the true value lies between 0"'06 and 0"'04, and we
probably should not place much confidence in any nearer
limits than 0"'07 and 0"'03. This is equivalent to saying
that the star is distant something between fourteen and
thirty-three parsecs from us. It will be seen that, when
the parallax is as low as 0"'05, even the best measures give
only the very roughest idea of the distance of the star,
and for smaller values the information becomes still more
vague. Clearly, to be of value a parallax must be at least
0"'05. It may be estimated that there are not more than
2000 stars so near as this, and a very large proportion of
these will be fainter than the tenth magnitude. The chances
are that, of five plates of the international Carte du Ciel
taken at random, only one will be fortunate enough to
pick up a serviceable parallax, and even that is likely to be
a very inconspicuous star, which would evade any but the
most thorough search. The prospect of so overwhelming a
proportion of negative results suggests that, for the present

1 6 STELLAR MOVEMENTS CHAP.

at any rate, work can be most usefully done on special
objects for which an exceptionally large proper motion
affords an a priori expectation of a sensible parallax. A
star of parallax 0"'05 may be expected to have a proper
motion of 20" per century or more, and that seems to be
a reasonable limit to work down to.

It will generally be admitted that a most valuable
extension of our knowledge will result from precise
measures of the distances of as many as possible of the
individual stars that come within the range above
mentioned. But many investigators have also sought to
determine the mean parallaxes of stars of different
magnitudes or motions. When the individual distances
are too uncertain, the means of a large number may still
have some significance. Whilst some useful results can be
and have been obtained by this kind of research, its
possibilities seem to be very limited. Generally speaking,
this class of determination requires even greater refine-
ment than the measurement of individual parallaxes ;
refinement which is scarcely yet within reach. For example,
the mean parallax of stars of the sixth magnitude is
0"*014 (perhaps a rather high estimate) ; that of the
comparison stars would probably be about half this, so that
the relative parallax actually measured would be 0"'007.
The possible systematic errors depending on magnitude
and colour (the mean colour of the sixth magnitude is
perhaps different from the ninth) make the problem of
determining this difference one of far higher difficulty than
that of measuring the parallax of an individual star. It
means gaining almost another decimal place beyond the
point yet reached. We need not dwell on the enormous
labour of observing the necessary fifty or one hundred
sixth magnitude stars to obtain this mean with reason-
able accuracy ; it might well be thought worth the
trouble ; but there is no evidence that systematic
errors have as yet been brought as low as 0" - 001

i THE DATA OF OBSERVATION 17

even in the best work, and indeed it seems almost
inconceivable.

From these considerations it appears that parallax-
determinations should be directed towards :
t (l) Individual stars with proper motions exceeding 20"
a century. This will yield many negative results, but a
fair proportion of successes.

(2) Classes of stars with proper motions less than 20",
but still much above the average. These parallaxes will
have to be found individually, but for the most part only
the mean result for a class will be of use.

(3) A possible extension to classes of stars not dis-
tinguished by large proper motion, provided it is realised
that a far higher standard is required for this work, and
that a freedom from systematic error as great as 0"*001 can
be ensured.

Proper Motion. --For stellar investigation the
proper motions, i.e.. the apparent angular motions of the
stars, form most valuable material. For extension in our
knowledge of magnitudes, parallaxes, radial velocities, and
spectral classification, we shall ultimately come to depend
on improved equipment and methods of observation ; but
the mere lapse of time enables the proper motions to
become known with greater and greater accuracy, and the
only limit to our knowledge is the labour that can be
devoted, and the number of centuries we are content to
wait.

Proper motions of stars differ greatly in amount, but in
general the motion of any reasonably bright star (e.g.,
brighter than 7 m '0) is large enough to be detected in the
time over which observations have already accumulated.
Whilst it is quite the exception for a star to have a
measurable parallax, it is exceptional for the proper motion
to be insensible. It may be useful to give some idea of
the certainty and trustworthiness of the proper motions in
ordinary use, though the figures are^ necessarily only

C

1 8 STELLAR MOVEMENTS CHAP.

approximate. When the probable error is about 1" per
century in both coordinates, the motion may be considered
to be determined fairly satisfactorily ; the Groombridge and
Carrington Catalogues, largely used in statistical investiga-
tions, are of about this order of accuracy. A higher
standard probable error about 0" 6 per century is
reached in Boss's "Preliminary General Catalogue of 6188
Stars," which is much the best source of proper motions
at present available. For some of the fundamental stars
regularly observed at a large number of places throughout
the last century the accidental error is as low as 0*'2 per
century ; but the inevitable systematic errors may well
make the true error somewhat larger. Various sources of
systematic error, particularly uncertainties in the constant
of precession and the motion of the equinox, may render
the motions in any region of the sky as much as 0"*5 per
century in error ; it is unlikely that the systematic error
of the best proper motions can be greater than this, except
in one or two special regions of southern declination,
where exceptional uncertainty exists.

"We may thus regard the proper motions used in
statistical researches as known with a probable error of
not more than 1" per century in right ascension and
declination. Koughly speaking, an average motion is from
3" to 7" per century. A centennial motion of more than
20" is considered large, although there are some stars which
greatly exceed this speed. The fastest of all is C.Z.
5 h 243 a star of the ninth magnitude found by J. C.
Kapteyn and R. T. A. Innes on the plates of the Cape
Photographic Durchmusterung which moves at the rate
of 870" per century. This speed would carry it over an
arc equal to the length of Orion's belt in just above a
thousand years. Table 1 shows the stars at present
known of which the centennial speed exceeds 300". The
number of faint stars on this list is very striking ; and, as
our information practically stops at the ninth magnitude, it

THE DATA OF OBSERVATION

may be conjectured that there are a number of still fainter
stars yet to be detected.

TABLE 1.
Stars with large Proper Motion.

Name. j&

Dec. nnual

1900. PP er
Motion.

Magnitude.

h. m.
C.Z. 5 h 243 5 8
Groombridge 1830 ... 11 47
Lacaille 9352 . . . 22 59

-45-0 870
+ 38-4 7-07
-36'4 7-02

8-3
6-5

7-4

Cordoba 32416
61 1 Cygni 21 2

-37-8 6-07
+ 38-3 5-25

8-5
5*6

Lalonde 21185 . 10 58

+ 36 '6 477

7 '6

f Indi . . . . 21 56

-57 -2 4-67

47

Lalonde 21258 . 11

+ 44-0 4-46

8-9

o 2 Eridani 4 11

- 7'8 4-08

4-5

fO.A. (s.) 14318 15 5
\O.A. (s.) 14320. 15 5

-16-0 3-76
-15-9 376

9-6
9'2

p. Cassiopeiae . . 12

+ 54-4 375

5'3

a 1 Centauri 14 33

-60-4 3*66

0*3

Lacaille 8760 ... . 21 11
e Eridani .... 3 16

-39-2 3-53
-43'4 3-15

7-3
4 '3

O.A. (x.) 11677 .... 11 15

+ 66-4 3-03.

9-2

Comparatively little is known of the motions of stars
fainter than the ninth magnitude. The Carrington proper
motions, discussed by F. W. Dyson, carry us down to
10 m '3 for the region within 9 of the North Pole. A
number of the larger proper motions of faint stars in the
Oxford Zone of the Carte du Ciel have been published by
H. H. Turner and F. A. Bellamy. 6 Further, by the reduc-
tion of micrometric measurements,^ G. C. Comstock 7 has
obtained a number of proper motions extending even to
the thirteenth magnitude. There is no difficulty to be
expected in securing data for faint stars ; but the work has
been taken up comparatively recently, and the one essential
is lapse of a sufficient time.

Radial Velocity. The velocity in the line of sight
is measured by means of a spectroscope. In accordance
with Doppler's Principle, the lines in the spectrum
of a star are displaced towards the red or the violet

c 2

20 STELLAR MOVEMENTS CHAP.

(relatively to a terrestrial comparison spectrum) according
as the star is receding from or approaching the earth.
Unlike the proper motion, the radial motion is found
directly in kilometres per second, so that the actual linear
speed, unmixed with the doubtful element of distance, is
known. Hitherto it has scarcely been possible to measure
the velocities of stars fainter than the fifth magnitude, but
that limitation is now being removed. The main difficulty
in regard to the use of the results is the large proportion
of spectroscopic binary stars, about one in three or four of
the total number observed. As the orbital motion is often
very much larger than the true radial velocity, it is
essential to allow sufficient time to elapse to detect any
variation in the motion, before assuming that the measures
give the real secular motion, of which we are in search.
Another uncertainty arises from possible systematic errors
affecting all the stars belonging to a particular type of
spectrum. There is reason to believe that the measured
velocity of recession of the Type B stars is systematically
5 km. per sec. too great. 8 This may be due to errors in
the standard wave-lengths employed, or to a pressure-shift
of the lines under the physical conditions prevailing in
this kind of star. Smaller errors affect the stars of other
types.

Apart from possible systematic error, a remarkable
accuracy has been attained in these observations. For a
star with sharp spectral lines a probable error of under
0'25 km. per sec. is well within reach. Stars of Types
B and A have more diffuse lines and the results are not
quite so good ; but the accuracy even in these cases is far
beyond the requirements of the statistician. The observed
velocities range up to above one hundred km. per sec., but
speeds greater than sixty km. per sec. are not very common.
The greatest speed yet measured is that of Lalande 19 GO,
viz. , 325 km. per sec. The next highest is C.Z. 5 h 243, already
mentioned as having the greatest apparent motion acr<>

i THE DATA OF OBSERVATION 21

the sky ; it is observed to be receding at the rate of
242 km. per sec., or 225 km. per sec. if we make
allowance for the Sun's own motion. As these figures
refer to only one component of motion, the total speeds of
stars are sometimes considerably greater.

Radial motions of about 1400 stars have now been
published, the great bulk of the observations having been
made at the Lick Observatory. Most of this material
only became accessible to investigators in 1913, and
there has scarcely yet been time to make full use of the
new data.

There are a few systems which can be observed both as
visual and as spectroscopic binaries. In such cases it is
possible to deduce the distance of the star by a method
quite independent of the usual parallax determinations.
From the visual observations, the period and the other
elements of the orbit can be found. The dimensions,
however, are all expressed in arc, i.e., in linear measure
divided by the unknown distance of the star. From these
elements we can calculate for any date the relative velocity
in the line of sight of the two components ; but this also
will be expressed as a linear velocity divided by the
unknown distance. By comparing this result with the
same relative velocity measured spectrographically, and
therefore directly in linear measure, the distance of the
system can be derived. This method is of very limited
application ; but in the case of a Centauri it has given a
very valuable confirmation of the parallax determined in
the ordinary way. It increases our confidence that the
usual method of measuring stellar distances is a sound one.

Mass and Density. Knowledge of the masses
and densities of stars is derived entirely from binary
systems. The sources of information are of three kinds :

(1) From visual binaries.

(2) From ordinary spectroscopic binaries.

(3) From eclipsing variables.

22

STELLAR MOVEMENTS

CHAP.

The combined mass of the two components of a binary
system can be found from the length of the major semi-
axis of the orbit a and the period P by the formula

Here the masses are expressed in terms of the Sun's mass
as unit, and the astronomical unit and the year are taken
as the units of length and time.

In a well-observed visual orbit, all the elements are
known (except for an ambiguity of sign of the inclination),
but the major axis is expressed in arc. This can be
converted into linear measure, if the parallax has been
determined ; and hence m^ + m 2 can be found. When further,
besides the relative orbit, a rough absolute orbit of one of
the components has been found, by meridian observations
or otherwise, the ratio f m l \ i m. 2 is determinate, and m 1 and
m 2 are deduced separately. Owing to the difficulty of
determining parallaxes, cases of a complete solution of this
kind are rare. They are, however, sufficient to indicate
the fact that the range in the masses of the stars is not at
all proportionate to the huge range in their luminosities.

TABLE 2.

Well-detei-mined Masses of Stars.

Combined System

Brighter Com-
ponent

Star

Mass
(Sun=l)

Period
Years

a

Parallax

Luminos-
ity
(Sun-1)

Spectral
Type

( Herculis . .

1-8

::i D 1-35

0-14

5-0

G

Procyon . . .

1-3

39-0

4-05

0-32

97

F5

Sirius ....

3"4

48-8

7-59

0-38

48-0

A

a Centauri . .

1-9

81-2 1771

0-76

2-0

G

7'i uphiuchi .

2-5

88-4

4-55

0-17

1-1

K

o 2 Eridani .

n-7

lHO-n 479

0-17

0-84

G

rj Cassiopeiae * .

J-0

328-0 9-48

0-20

1-4

F5

8

* Another published orbit P = 508y -12'2" gives the mass =0'9. The
great uncertainty of the orbit appears to have little effect on the result.

i THE DATA OF OBSERVATION 23

Table 2 contains all the systems, of which the masses can be
ascertained with reasonable accuracy, i.e., systems for which
good orbits 9 and good parallax-determinations 10 have been
published. Possibly some of the more doubtful orbits
would have been good enough for the purpose, but I doubt
if the list could be much extended without lowering the
standard.

Another fact which appears is that the ratio of the
masses of the two components of a binary is generally not
far from equality, notwithstanding considerable differences
in the luminosity. Thus Lewis Boss 11 in ten systems found
that the ratio of the mass of the faint star to the brighter
star ranged from 0'33 to IT,* the mean being 071. The
result is confirmed by observations of such spectroscopic
binaries as show the lines of both components, though in
this case the disparity of luminosity cannot be so great.

Even w r hen the parallax is not known, important
information as to the density can be obtained. Consider
for simplicity a system in which one component is of
negligible mass ; the application to the more general case
requires only slight modifications, provided mjm. 2 is known
or can be assumed to have its average value.
Let

d be the distance of the star

6 its radius

S its surface brightness

L, I its intrinsic and apparent luminosities

M its mass

p its density

y the constant of gravitation

Then

L = nV

and

* Excluding one very doubtful result.

24 STELLAR MOVEMENTS CHAP.

From these

- 7 is the semi-axis of the orbit in arc, and I and P are
a

observed quantities ; consequently the coefficient of S% is
known. We have thus an expression for the density in
terms of the surface brightness, and can at least compare
the densities of those stars which, on spectroscopic evi-
dence, may be presumed to have similar surface conditions.

The deusity is found to have a large range, many of the
stars being apparently in a very diffused state with
densities perhaps not greater than that of atmospheric air.

The spectroscopic binaries also give some information as
to the masses of stars. The formula (m x + m 2 ) = a 3 /P 2 is
applicable, and as a is now found in linear measure it is
not necessary to know the parallax. The quantity deduced
from the observations is, however, in this case a sin i,
where i is the inclination of the plane of the orbit. The
inclination remains unknown, except when the star is an
eclipsing variable * or in the rare case when the system
is at the same time a visual and a spectroscopic binary.
For statistical purposes, such as comparing the masses of
different types of stars, we may assume that in the mean
of a sufficient number of cases the planes of the orbits will
be distributed at random, and can adopt a mean value for
sin i. Thus from spectroscopic binaries the average masses
of classes, but not of individual stars, can be found.

In the case of eclipsing variable stars, the densities of
the two components can be deduced entirely from the light-
curve of the star. Although these are necessarily spectro-
scopic binary systems, observations of the radial velocity
are not needed, and are not used in the results. The
actual procedure, which is due to H. N. Russell and
H. Shapley, 12 is too complicated to be detailed here, as

* In this case it is evident that * must be nearly 90, and accordingly sin i
may be taken as unity.

i THE DATA OF OBSERVATION 25

it bears only incidentally on our subject ; but the general
principle may be briefly indicated, it will be easily
realised that the proportionate duration of the eclipse and
other features of the light curve do not depend on the
absolute dimensions of the system, but on the ratio of the
three linear quantities involved, viz., the diameters of the
two stars and the distance between their centres. By
strictly geometrical considerations therefore, we find the
radii r lt r 2 of the stars expressed in terms of the unknown
semi-axis of the orbit a as unit. Now the relation between
the mass and density of a star involves the cube of the
radius, and the dynamical relation between the mass and
the period involves the cube of a. Thus, on division, the
absolute masses and the unknown unit a disappear simul-
taneously, and we are left with the density expressed in

7* /T

terms of the period and the known ratios , . Tiie

a a

key to the solution is that in astronomical units the
Dimensions of density are (time)" 2 ; the density thus
depends on the period, and on the ratios, but not the
absolute values, of the other constants of the system.

The densities found in this way are not quite rigorously
determined. It is necessary to assume a value of the
ratio of the masses of the two stars ; as already explained,
this ratio does not differ widely from unity, but in extreme
cases the results may be as much as fifty per cent, in error
from this cause. Further, the darkening at the limb of
the star has some effect on the determination, and the
assumed law of darkening is hypothetical. By taking
different assumptions, between which the truth is bound
to lie, it can be shown that these uncertainties do not
amount to anything important, when regard is had to the
great range in stellar densities which is actually found.

We may conclude this account of the nature of the
observations, on which our knowledge of the stellar

26 STELLAR MOVEMENTS CHAP.

universe is based, by a reference to J. C. Kapteyn's " Plan
of Selected Areas." When the study of stars was confined
mainly to those brighter than the seventh magnitude, and
again when it was extended as far as the ninth, or tenth,
a complete survey of all the stars was not an impossible
aim, and indeed all the data obtainable could well be
utilised. But investigations are now being pushed towards
the fifteenth and even lower magnitudes. These fainter
stars are so numerous that it is impossible and unneces-
sary to do more than make a selection. As the different
kinds of observation for parallax, proper motion, magnitude,
spectral type, and radial velocity are highly specialised and
usually carried out at different observatories, some co-opera-
tion is necessary in order that so far as possible the
observations may be concentrated on the same groups of
stars. Kapteyn's 13 plan of devoting attention to 206
selected areas, distributed all over the sky, so as to
cover all varieties of stellar distribution, has met with
very general support. The areas have their centres on or
near the circles of declination, 0, 15, 30, 45,
=t 60, 75, 90. The exact centres have been chosen
with regard to various practical considerations ; but the
distribution is very nearly uniform. In addition to the
main " Plan," 46 areas in the Milky Way have been
chosen, typical of its main varieties of structure. The
area proper consists of a square 75' x 75', or alternatively
a circle of 42' radius ; but the dimensions may be extended
or diminished for investigations of particular data.

The whole scheme of work includes nine main sub-
divisions : (1) A Durchmusterung of the areas. (2)
Standard photographic magnitudes. (3) Visual and
photovisual magnitudes. (4) Parallaxes. (5) Differential
proper motions. (6) Standard proper motions. (7) Spectra.
(8) Radial velocities. (9) Intensity of the background of
the sky. The Durchmusterung is well advanced ; it will
include all stars to 17 m , the positions being given with a

i THE DATA OF OBSERVATION 27

probable error of about 1" in each co-ordinate, and the
magnitudes (differential so far as this part of the work is
concerned) with a probable error of O m *l. Considerable
progress has been made with the determination of sequences
of standard photographic magnitudes for each area. The
work of determining the visual magnitudes has been partly
accomplished for the northern zones. For parallaxes, most
of the areas have been portioned out between different
Observatories ; the greatest progress has been made at the
Cape Observatory for the southern sky, but the plates
have not yet been measured. From what has already
been stated with regard to the practical possibilities of
parallax-determinations, it will be seen that there is some
doubt as to the utility of this part of the Plan. For the
proper motions of faint stars, the work has necessarily been
confined mainly to obtaining plates for the initial epoch. At
the Radcliffe Observatory, 150 plates have been stored away
undeveloped, ready for a second exposure after a suitable
interval, but in most cases it is intended to rely on the
parallax plates for giving the initial positions. For standard
proper motions in the northern sky, observations are
shortly to be started at Bonn ; these will serve for com-
parison with older catalogues, but they may also be regarded
as initial observations for more accurate determinations
in the future. Determinations of spectral type as far as
the ninth magnitude, made at Harvard, will shortly be
available for these areas and, indeed, for the whole sky.
The extension to the eleventh magnitude is very desirable,
and is one of the most urgent problems of the whole
Plan. Radial velocity determinations are being pressed
as far as 8 m *0 at Mount Wilson, but rapid progress
is not to be expected until the completion of the 100-
inch reflector. A valuable, though unofficial, addition
to the programme is E. A. Fath's Durchmusterung u of
all the nebulae in the areas from the North pole to
Dec. -15.

28 STELLAR MOVEMENTS CHAP.

REFERENCES. CHAPTER I.

1. Ludendorff, Astr. Nach., No. 4547.

2. Hertzsprung, Astr. Nach., No. 4296.

3. King, Harvard Annalx, Vol. 59, p. 152.

4. Schwarzschild, " Aktinometrie, " Teil B, p. 19.

5. Hertzsprung, Potsdam Publications, No. 63 ; Astr. Nach., No. 4362
(contains a bibliography).

6. Monthly Notices, Vol. 74, p. 26.

7. Comstock, Pub. Washburn Observatory, Vol. 12, Pt. 1.

8. Campbell, Lick Bulletin, No. 195, p. 104.

9. Aitken, Lick Bulletin, No. 84.

10. Kapteyn and Weersma, Groningen Publications, No. 24.

11. Boss, Preliminary General Catalogue of 6188 Stars, Introduction,

p. 23.

12. Russell and Shapley, Astrophysical Journal, Vol. 35, p. 315, et seq.

13. Kapteyn, "Plan of Selected Areas"; ditto, ''First and Second

Reports " ; Monthly Notices, Vol. 74, p. 348.

14. Fath, Astronomical Journal, Nos. 658-9.

BIBLIOGRAPHY.

Magnitudes. The Harvard Standard Polar Sequence is given in Harvard
Circular, No. 170. For an examination by Scares, see Astrophysical Journal,
Vol. 38, p. 241.

The chief catalogues of visual stellar magnitudes are :
** Revised Photometry," Harv. Ann., Vols. 50 and 54.
Miiller and Kempf, Potsdam Publications, Vol. 17.

For photographic magnitudes :

Schwarzschild, "Aktinometrie," Teil B (Gottingen Abhandlungen,

Vol. 8, No. 4).

Greenurich Astrographic Catalogue, Vol. 3 (advance section).
Parkhurst, Astrophysical Journal, Vol. 36, p. 169.

A useful discussion of the methods of determining photographic magni-
tude will be found in an R.A.S. " Council Note," Monthly Notices,
Vol. 73, p. 291 (1913).

Spectral Type*. The most extensive determinations are to be found in
Jltirv. Ann., Vol. 50, which gives the type for stars brighter than 6 m> 5.
Scattered determinations of many fainter stars also exist. It is understood
that a very comprehensive catalogue containing the types of 200,000 stars
will shortly be issued from Harvard.

A description of the principles of the Draper classification is given in
Harv. Ann., Vol. 28, pp. 140, 146.

Parallaxes. The principal sources are :

Peter, Abhandlunyen kb'niglichsiichsische Gesell. der Wissenschaften,

Vol. 22, p. 239, and Vol. 24, p. 179.
Gill, Cape Annals, Vol. 8, Pt. 2 (1900).

i THE DATA OF OBSERVATION 29

Schlesinger, Aatrophyaical Journal, Vol. 34, p. 28.
Russell and Hinks, Astronomical Journal, No. 618-9.
Elkin, Chase and Smith, Yale Transactions, Vol. 2, p. 389.
Slocum and Mitchell, J-sfro^/j^/m^ Jmrrwd, Vol.38, p. 1.

Very useful compilations of the parallaxes taken from all available sources
are given by Kapteyn and Weersma, Qroningen Publications, No. 24, and
by Bigourdan, Bulletin Axtronmiiiiiue, Vol. 26. Fuller references are given
in these publications.

Proper Motions. Lewis Boss's Preliminary General Catalogue of 6188
Stars, which includes proper motions of all the brighter stars, supersedes
many earlier collections.

Dyson and Thackeray's New Reduction of Groombridge's Catalogue contains
4243 stars, including many fainter than the eighth magnitude, within
50' of the North Pole.

Still fainter stars are contained in the Greenwich-Carrington proper
motions discussed by Dyson. Some of these are published in the Second
Nine-Year Catalogue (1900), but certain additional corrections are required
to the motions there given (see Monthly Notices, Vol. 73, p. 336). The
complete list (unpublished) contains 3735 stars.

Porter's Catalogue, Cincinnati Publications. No. 12, gives 1340 stars
of especially large proper motion.

Radial Velocities. Catalogues containing a practically complete summary
of the radial velocities at present available for discussion are given by
Campbell in Lick Bulletin, Nos. 195, 211, and 229. These contain about
1350 stars.

CHAPTER II

GENERAL OUTLINE

THIS chapter will be devoted to a general description of
the sidereal universe as it is revealed by modern researches.
The evidence for the statements now made will appear
gradually in the subsequent part of the book, and minor
details will be filled in. But it seems necessary to
presume a general acquaintance with the whole field of
knowledge before starting on any one line of detailed
investigation. At first sight it might seem possible to
divide the subject into compartments the distribution of
the stars through space, their luminosities, their motions,
and the characters of the different spectral types but it is
not possible to pursue these different branches of inquiry
independently. Any one mode of investigation leads, as a
rule, to results in which all these matters are involved
together, and no one inquiry can be worked out to a
conclusion without frequent reference to parallel investiga-
tions. We have therefore adopted the unusual course of
placing what may be regarded as a summary thus early in
the book.

In presenting a summary, we may claim the privilege of
neglecting many awkward difficulties and uncertainties
that arise, promising to deal fairly with them later. We
can pass over alternative explanations, which for the
moment are out of favour ; though they need to be kept

30

CH. ii GENERAL OUTLINE 31

alive, for at any moment new facts may be found, which
will cause us to turn to them again. The bare outline,
devoid of the details, must not be taken as an adequate
presentation of our knowledge, and in particular it will
fail to convey the real complexity of the phenomena
discussed. Above all, let it be remembered that our object
in building up a connected idea of the universe from the
facts of observation is not to assert as unalterable truth
the views we arrive at, but, by means of working hypotheses,
to assist the mind to grasp the interrelations of the facts,
and to prepare the way for a further advance. When we
look back on the many transformations that theories in all
departments of science have undergone in the past, we

m =*:

-Galactic
Plane

FIG. 1. Hypothetical Sect on of the Stellar System.

shall not be so rash as to suppose that the mystery of the
sidereal universe has yielded almost at the first attack.
But as each revolution of thought has contained some
kernel of surviving truth, so we may hope that our
present representation of the universe contains something
that will last, notwithstanding its faulty expression.

It is believed that the great mass of the stars with
which we are concerned in these researches are arranged
in the form of a lens- or bun-shaped system. That is to
say, the system is considerably flattened towards one plane.
A general idea of the arrangement is given in Fig. 1 , where
the middle patch represents the system to which we are
now referring. In this aggregation the Sun occupies a fairly
central position, indicated by -K The median plane of the
lens is the same as the plane marked out in the sky by the

32 STELLAR MOVEMENTS CHAP.

Milky Way, so that, when we look in any direction along
the galactic plane (as the plane of the Milky Way is called),
we are looking towards the perimeter of the lens where the
boundary is most remote. At right angles to this, that is,
towards the north and south galactic poles, the boundary is
nearest to us ; so near, indeed, that our telescopes can
penetrate to its limits. The actual position of the Sun is
a little north of the median plane ; there is little evidence
as to its position with respect to the perimeter of the
lens ; all that we can say is that it is not markedly
eccentric.

The thickness of the system, though enormous com-
pared with ordinary units, is not immeasurably great. No
definite distance can be specified, because it is unlikely
that there is a sharp boundary ; there is only a gradual
thinning out of the stars. The facts would perhaps be
best expressed by saying that the surfaces of equal density
resemble oblate spheroids. To give a general idea of the
scale of the system, it may be stated that in directions
towards the galactic poles the density continues practically
uniform up to a distance of about 100 parsecs ; after that
the falling off becomes noticeable, so that at 300 parsecs it
is only a fraction (perhaps a fifth) of the density near the
Sun. The extension in the galactic plane is at least three
times greater. These figures are subject to large
uncertainties.

It seems that near the Sun the stars are scattered in a
fairly uniform manner ; any irregularities are on a small
scale, and may be overlooked in considering the general
architecture of the stellar system. But in the remoter
parts of the lens, or more probably right beyond it, there
lies the great cluster or series of star-clouds which make
up the Milky Way. In Fig. 1 this is indicated (in section)
by the star-groups to the extreme left and right. These
star-clouds form a belt stretching completely round the
main flattened system, a series of irregular agglomera-

ii GENERAL OUTLINE 33

tions of stars of wonderful richness, diverse in form and
grouping, but keeping close to the fundamental plane.
It is important to distinguish clearly the two properties of
the galactic plane, for they have sometimes been con-
fused. First, it is the median plane of the bun-shaped
arrangement of the nearer stars, and, secondly, it is the
plane in which the star-clouds of the Milky Way are coiled.

Not all the stars are equally condensed to the galactic
plane. Generally speaking, the stars of early type con-
gregate there strongly, whereas those of late types are
distributed in a much less flattened, or even in a practically
globular, form. The mean result is a decidedly oblate
system ; but if, for example, we consider separately the
stars of Type M, many of which are at a great distance
from us, they appear to form a nearly spherical system.

In the Milky Way are found some vast tracts of absorbing
matter, which cut off the light of the stars behind.
These are of the same nature as the extended irregular
nebulae, which are also generally associated with the
Milky Way. The dark absorbing patches and the faintly
shining nebulae fade into one another insensibly, so
that we may have a dark region with a faintly luminous
edge. Whether the material is faintly luminous or not,
it exercises the same effect in dimming or hiding the
bodies behind it. There is probably some of this absorb-
ing stuff even within the limits of the central aggregation.
In addition to these specially opaque regions, it is probable
that fine particles may be diffused generally through
interstellar space, which would have the effect of dimming
the light of the more distant stars ; but, so far as can be
ascertained, this " fog " is not sufficient to produce any
important effect, and we shall usually neglect it in the
investigations which follow.

In studying the movements of the stars we necessarily
leave the remoter parts of space, confining attention mainly
to the lens-shaped system, and perhaps only to the inner

D

34. STELLAR MOVEMENTS CHAP.

parts of it, where the apparent angular movements are
appreciable. Researches on radial motions need, not be
quite so limited, because in them the quantity to be
measured is independent of the distance of the stars ;
but here too the nearer parts of the system obtain a
preference, for observations are confined to the bright
stars. Although thus restricted, our sphere of knowledge
is yet wide enough to embrace some hundreds of thousands
of stars (considered through representative samples) ; the
results that are deduced will have a more than local
importance.

The remarkable result appears that within the inner
system the stars move with a strong preference in two
opposite directions in the galactic plane. There are
two favoured directions of motion ; and the appearance
is as though two large aggregates of stars of more or less
independent origin were passing through one another, and
so for the time being were intermingled. It is true that
such a straightforward interpretation seems to be at
variance with the plan of a single oblate system, which
has just been sketched. Various alternatives will be
considered later ; meanwhile it is sufficient to note that
the difficulty exists. But, whatever may be the physical
cause, there is no doubt that one line in the galactic
plane is singled out, and the stars tend to move to and
fro along it in preference to any transverse directions.
We shall find it convenient to distinguish the two streams
of stars, which move in opposite directions along the
line, reserving judgment as to whether they are really
two independent systems or whether there is some other
origin for this curious phenomenon. The names assigned
are

Stream I. moving towards R. A. 94, Dec. +12'.
II. ,, R.A. 274% Dec. -12.

The relative motion of one stream with respect to
the other is about 40 km. per sec.

ii GENERAL OUTLINE 35

The Sun itself has an individual motion with respect
to the mean of all the stars. Its velocity is 20 kilo-
metres per second directed towards the point R.A. 270
Dec. + 35. The stellar movements that are directly
observed are referred to the Sun as standard, and are
consequently affected by its motion. This makes a
considerable alteration in the apparent directions of the
two streams ; thus we find

Stream I. moving towards R.A. 91 C , Dec. - 15 ^ relatively to
II. R.A. 288, Dec. -64 / the Sun ;

and moreover the velocity of the first stream is about 1'8
times that of the second (probably 34 and 19 km. per sec.,
respectively). Stream I is sometimes therefore referred to
as the quick-moving stream, and Stream II as the slow-
moving one ; but it must be remembered that this
description refers only to the motion relative to the
Sun. The stars which constitute the streams have,
besides the stream-motion, individual motions of their
own ; but the stream-motion sufficiently dominates over
these random motions to cause a marked general agreement
of direction.

Stream I contains more stars than Stream II in the
ratio 3 : 2. Though this ratio varies irregularly in different
parts of the sky, the mixture is everywhere fairly
complete. Moreover, there is no appreciable difference in
the average distances of the stars of the two streams. It
is not a case of a group of nearer stars moving in one
direction across a background of stars moving the opposite
way ; there is evidence that the two streams thoroughly
permeate each other at all distances and in all parts of the
heavens.

A more minute investigation of this phenomenon shows
that it is complicated by differences in the behaviour of
stars according to their spectral type. An analysis which
treats the heterogeneous mass of the stars as a whole

D 2

36 STELLAR MOVEMENTS CHAP.

without any separation of the different types will fail to
give a complete insight into the phenomenon. But, until
a great deal more material is accumulated, this interrelation
of stream motions and spectral type cannot be worked out
very satisfactorily. The outstanding feature, however, is
that the stars of the Orion Type (Type B) seem riot to
share to any appreciable extent in the star-streaming
tendency. Their individual motions, which are always
very small, are nearly haphazard, though the apparent
motions are, of course, affected by the solar motion. They
thus form a third system, having the motion of neither of
the two great streams, but nearly at rest relatively to the
mean of the stars. This third system is not entirely
confined to the B stars. In the ordinary analysis into
two streams we always find some stars left over com-
paratively few in number yet constituting a distinct
irregularity which evidently belong to the same system.
These stars may be of any of the spectral types. There is
something arbitrary in this dissection into streams (which
may be compared to a Fourier or spherical harmonic
analysis of observations), and we can, if we like, adopt a
dissection which gives much fuller recognition to this
third system.

At one time it seemed that the third stream, Stream
as it is called, might be constituted of the very distai.t
stars, lying beyond those whose motions are the main
theme of discussion. If that were so, it would not be
surprising to find that they followed a different law, and
were not comprised in the two main streams. But this
explanation is now found to be at variance with the facts.
\\ have to recognise that Stream is to be found even
among the nearer stars.

The smallness of the individual movements of the
B stars is found to be part of a much more general law.
Astrophysicists have by a study of the spectra arranged
the stars in what they believe to be the successive stages

ii GENERAL OUTLINE 37

of evolution. Now it is found that there is a regular
progression in the size of the linear motions from the
youngest to the oldest stars. It is as though a star was
born without motion, and gradually acquires or grows one.
The average individual motion (resolved in one direction)
increases steadily from about 6*5 km. per sec. for Type B
to 1 7 km. per sec. for Type M.

We may well regard this relation of age and velocity as
one of the most startling results of modern astronomy.
For the last forty years astrophysicists have been studying
the spectra and arranging the stars in order of evolution.
However plausible may be their arguments one would have
said that their hypotheses must be for ever outside the
possibility of confirmation. Yet, if this result is right,
we have a totally distinct criterion by which the stars are
arranged in the same order. If it is really true that the
mean motion of a class of stars measures its progress along
the path of evolution, we have a new and powerful aid to
the understanding of the steps of stellar development.

It is not at all easy to explain why the stellar velocities
increase with advancing development. I am inclined to
think that the following hypothesis offers the best
explanation of the facts. In a primitive state the star-
forming material was scattered much as the stars are now,
that is, densely along the galactic plane up to moderate
distances, and more thinly away from the plane and at
great distances. Where the material was rich, large stars,
which evolved slowly, were formed ; where it was rare,
small stars, which developed rapidly. The former are our
early type stars ; not having fallen in from any great
distance they move slowly and in the main parallel to the
galactic plane. The latter our late type stars have
been formed at a great distance, and have acquired large
velocities in falling in ; moreover since they were not
necessarily formed near the galactic plane, their motions
are not so predominantly parallel to it.

38 STELLAR MOVEMENTS CHAP.

Whilst the individual motion of a star gradually increases
from type to type, the stream- motion appears with remark-
able suddenness. Throughout Type B, even up to B 8 and
B 9, the two star-streams are unperceived ; but in the next
type, A, the phenomenon is seen in its clearest and most
pronounced form. Through the remaining types it is still
very prominent, but there is an appreciable falling off.
There is no reason to believe that this decline is due to
any actual decrease in the stream-velocities ; it is only that
the gradual increase of the haphazard motions renders the
systematic motions less dominant.

Among the most beautiful objects that the telescope
reveals are the star-clusters, particularly the globular
clusters, in which hundreds or even thousands of stars are
crowded into a compact mass easily comprised within the
field of a telescope. Recent research has revealed several
systems, presumably of a similar nature to these, which
are actually in our neighbourhood, and in one case even
surrounding us. Being seen from a short distance the
concentration is lost, and the cluster scarcely attracts notice.
The detection of these systems relatively close to us is an
important branch of study ; they are distinguished by the
members having all precisely equal and parallel motions.
The stars seem to be at quite ordinary stellar distances
apart, and their mutual at traction is too weak to cause any
appreciable orbital motion. They are not held together by
any force ; and we can only infer that they continue to
move together because no force has ever intervened to
separate them.

These " moving clusters" are contained within the central
aggregation of stars. Many of the globular clusters,
though much more distant, are probably also contained in
it ; others, however, may be situated in the star-clouds of
the Milky Way. Their distribution in the sky is curiously
uneven ; they are nearly all contained in one hemisphere.
They are most abundant in Sagittarius and Ophiuchus,

ii GENERAL OUTLINE 39

near a brilliant patch of the Milky Way, which is
undoubtedly the most extraordinary region in the sky.
This might be described as the home of the globular
clusters.

We shall also have to consider the nebulae, and their
relation to the system of the stars. At this stage it
may be sufficient to state that under the name "nebulae"
are grouped together a number of objects of widely
differing constitution ; we must not be deceived into
supposing that the different species have anything in
common. There is some reason for thinking that the spiral
or "white" nebulas are objects actually outside the whole
stellar system, that they are indeed stellar systems coequal
with our own, and isolated from us by a vast intervening
void. But the gaseous irregular nebulae, and probably
also the planetary nebulae, are more closely associated with
the stars and must be placed among them.

It is now time to turn from this outline of the leading-
phenomena to a more detailed consideration of the
problems. The procedure will be to treat first the nearest
stars, of which our knowledge is unusually full and direct.
From these we pass to other groups that happen to be
specially instructive. From this very limited number of
stars, a certain amount of generalisation is permissible,
but our next duty is to consider the motions of the stars
in general ; this will occupy Chapters V. VII. After
considering the dependence of the various phenomena on
spectral type, we pass on to the problems of stellar
distribution. This comes after our treatment of stellar
motions, because the proper motions are, when carefully
treated, among the most important sources of information
as to the distances of stars. In Chapter XL we pass to
subjects the Milky Way and Nebulae of which our
knowledge is even more indefinite. The concluding Chapter
attempts to introduce the problem of the dynamical forces
under which motions of the stellar system are maintained.

CHAPTER III

THE NEAREST STARS

MOST of our knowledge of the distribution of the stars
is derived by indirect methods. Statistics of stellar magni-
tudes and motions are analysed and inferences are drawn
from them. In this Chapter, however, we shall consider
what may be learnt from those stars the distances of which
have been measured directly ; and, although but a small
sample of the stellar system comes under review in this
way, it forms an excellent starting point, from which
we may proceed to investigations that are usually more
hypothetical in their basis.

For a parallax-determination of the highest order of
accuracy, the probable error is usually about 0"'01. Thus
the position of a star in space is subject to a compara-
tively large uncertainty, unless its parallax amounts to at
least a tenth of a second of arc. How small a ratio such
stars bear to the whole number may be judged from the
fact that the median parallax of the stars visible to the
naked eye is only 0"'008 ; as many naked-eye stars have
parallaxes below this figure as above it. We are therefore
in this Chapter confined to the merest fringe of the
surrounding universe, making for the present no attempt
to penetrate into the general mass of the stars.

Table 3 shows all the stars that have been found to have

40

CH. Ill

THE NEAREST STARS

parallaxes of 0"*20 or greater.* Only the most trustworthy
determinations have been accepted, and in most cases at least
two independent investigators have confirmed one another.
The list is based mainly on Kapteyn and Weersma's com-
pilation. 1

TABLE 3.

The Nineteen Nearest Stars.
(Stars distant less than five parsecs from the Sun.)

Star.

Magni-
tude.

Spectrum.

Parallax.

Luminos-
ity

(Sun = l).

Remarks.

Groombridge 34 .
TJ Cassiopeiae . . .
T Ceti

8-2
3-6
3-6

Ma

F8
K

0-28
0-20
0-33-

o-oio

1-4
0-50

Binary
Binary

f Eridani . ...
CZ5 h 243 . . .
Sirius

3-3
8-3
- 1-6

K

G-K

A

0-31
0-32

0'38

0-79
0-007
48-0

Binary

Procyon . ...
Lai. 21185 . . .
Lai. 21258 . . .
OA (N.) 11677 . .

a Centauri ....

A (N.) 17415 . .
Pos. Med. 2164 . .
v Draconis ....
a Aquilne ....
61 Cygni
f Indi

0-5
7-6
8-9
9-2

0-3

9-3

8-8
4-8
0-9
5-6

4'7

F5
Ma
Ma

G, K5

F
K

-rr

A5
K5
K5

0-32-
0-40-
0-20
0-20

0-76'

0-27
0-29
0-20
0-24
0-31
0-28

97
0-009

0-011

0-008
J2-0 |
10-6 J
0-004
0-006
0-5
12-3

o-io

0-25

Binary

Binary
Binary

Binary

Kriiger 60 ....
Lacaille 9352 . . .

9-2

7'4

Ma

0-26
0-29

0-005
0-019

Binary

It is of much interest to inquire how far this list of
nineteen stars is exhaustive. Does it include all the stars
in a sphere about the Sun as centre with radius five
parsecs ? In one respect the Table is admittedly incom-
plete ; for stars fainter than magnitude 9'5 (on the B.D.
scale), determinations are entirely lacking. A 9" U 5 star
with a parallax 0"'2 would have a luminosity O'OOG, the

* In using a table of this kind I am following the Astronomer Royal,
F. \\ . Dyson, who first showed me its importance.

42 STELLAR MOVEMENTS CHAP.

Sun being the unit ; so that, in general, stars giving less
than l/200th of the light of the Sun could not be
included in the list. The distribution of the luminosities
in column five of the Table leads us to expect that these
very feeble stars may be rather numerous.

Admitting, then, that Table 3 breaks off at about
luminosity O'OOG, and that in all probability numerous
fainter stars exist within the sphere, how far is it complete
above this limit ? Generally speaking, stars are selected
for parallax-determinations on account of their large
proper motions. Most of the very bright stars have also
been measured, but in no case has a parallax greater than
0"'2 been found which was not already rendered probable
by the existence of a large proper motion. To form some
idea of the completeness with which the stars have been
surveyed for parallax, consider those stars the motions of
which exceed 1" per annum. F. W. Dyson has given a list
of ninety-five of these stars,' 2 and it is probable that his list
is nearly complete, at least as far as the ninth magnitude ;
the Durchmusterungs and Meridian Catalogues of most
parts of the sky have involved so thorough a scrutiny,
that it would be difficult for motions as large as this to
remain unnoticed. Of these ninety-five stars, sixty-five
may be considered to have well-determined parallaxes, or
at least the determinations have sufficed to show that they
lie beyond the limits of our sphere ; among the former are
seventeen of the nineteen stars of Table 3. For the
remaining thirty, either measurements have been unat-
tempted, or the determinations do not negative the possi-
bility of their falling within the sphere. This remainder
is not likely to be sa rich in large parallaxes, because it
includes a rather large proportion of stars which only just
exceed the annual motion of 1" ; but there are some
notable exceptions. The star Cordoba 32416, mag. 8*5,
having the enormous annual motion of G"'07 seems to
have been left alone entirely. It may be expected that

in THE NEAREST STARS 43

further examination of these thirty stars will yield four or
five additional members for our Table.

Of stars with annual proper motions less than 1" the
Table contains only two. It is not difficult to show that
this is an inadequate proportion. The median parallax of
stars distributed uniformly through the sphere (of radius
5 parsecs) is 0"'25 ; now for a star of that parallax an
annual motion of 1" would be equivalent to a linear
transverse motion of 20 km. per sec. Approximately
then for our sphere

No. of PM's > 1" No. of transverse motions > 20 km. /sec.

No. of PM's < 1" No. of transverse motions < 20 km. /sec.

Now our general knowledge of stellar velocities, derived
from other sources, is probably sufficiently good to give
a rough idea of the latter ratio ; for it may be expected
that the distribution of linear velocities within the sphere
will not differ much from the general distribution outside.
Taking the average radial velocity of a star as 17 km.
per sec. (the figure given by Campbell for types K and M,
which constitute the great majority of the stars), a
Maxwellian distribution would give 64 per cent, of the
transverse motions greater than 20 km. per sec. and 36
per cent, less a ratio of 1*8 : 1. The addition of the
solar motion will increase the proportion of high velocities ;
and in those parts of the sky where it has its full effect
the ratio is nearly 3:1. Probably we shall not be far
wrong in assuming that there will be two-fifths as many
stars with motions below 1" per annum as above it.

Having regard to these considerations, the calculation
stands thus

No. of stars in the Table with proper motion greater than 1" . 17

Proportionate allowance for stars not yet examined 5

Due proportion with proper motion less than 1", say .... 9

The Sun 1

Total 32

44 STELLAR MOVEMENTS CHAP.

To this must be added an unknown but probably consider-
able number of stars the luminosity of which is less than
1 /200th of the Sun.

In round numbers we shall take thirty as the density
of the stars within the sphere (tacitly ignoring the
intrinsically faint stars). As twenty of these are actually
identified, the number may be considered to rest on
observation with very little assistance from hypothetical
considerations.

This short list of the nearest stars well repays a careful
study. Many of the leading facts of stellar distribution
are contained in it; and, although it would be unsafe to
generalise from so small a sample, results are suggested
that may.be verified by more extensive studies.

Perhaps the most striking feature is the number of
double stars. It will be seen that eight out of the nine-
teen are marked " binary." Why some stars have split
into two components, whilst others have held together, is
an interesting question ; but it appears that the fission of
a star is by no means an abnormal fate. The stars which
separate into two appear to be not much less numerous
than those which remain intact. The large number of
discoveries of variable radial velocity made with the
spectroscope confirms this inference, though the spectro-
scopists generally do not give quite so high a proportion.
\V. W. Campbell 3 from an examination of 1600 stars
concludes that one-quarter are spectroscopic binaries. But
this proportion must be increased if visual binaries are
included (for these are not usually revealed by the spectro-
scope) ; and, in addition, there must be pairs too far
separated to be detected as spectroscopic binaries, but too
distant from us to be recognised visually. E. B. Frost,
examining the stars of Type B, found that two-fifths of
those on his programme were binary ; he also found that
in Boss's Taurus cluster the proportion was one-half. 4

in THE NEAREST STARS 45

In the Ursa Major cluster nine stars out of fifteen are
known to be binary.' Apparently the division into two
bodies takes place at a very early stage in a star's
history or in the pre-stellar state, as is evidenced by the
high proportion found in the earliest spectral type. As
time passes the components separate further from each
other and the orbital velocity becomes small, so that in the
later types an increasing proportion escapes detection.
There thus seems no reason to doubt that the proportion
eight out of nineteen very fairly represents the general
average, but at the very lowest it cannot be less than one
in three.

The luminosities of the stars in the Table range from
48 to 0*004, that of the Sun being taken as unit. We
have already seen that the lower limit is due to the fact
that our information breaks off near this point, and it is
natural to expect that there must be a continuous series of
fainter bodies terminating with totally extinct stars. At
the other end of the scale a larger sample would un-
doubtedly contain stars of much greater luminosity ; but
these are comparatively rare in space. Arcturus, for
example, is from 150 to 350 times as bright as the Sun,
An tares at least 180 times, whilst Eigel and Canopus can
scarcely be less than 2000 times as bright, allowing in
each case a wide margin for the possible uncertainty of the
measured parallaxes. There is little doubt that these
estimates err on the side of excessive caution.

For a star of the same intrinsic luminosity as the Sun
to appear as bright as the sixth magnitude, its parallax
must be not less than 0"'08. Since there is no doubt that
the majority of stars visible to the naked eye are much
further away than this, it follows that the great majority
of these stars must be much brighter, in fact more than a
hundred times as bright as the Sun. We might hastily
suppose that the Sun is therefore far below the average
brilliancy. But Table 3 reveals a very different state of

4 6 STELLAR MOVEMENTS CHAP.

things. Of the nineteen stars, only five exceed it, whilst
fourteen are fainter. The apparent paradox directs attention
to a fact which we shall have occasion to notice frequently.
The stars visible to the naked eye, and the stars
enumerated in the catalogues, are quite unrepresentative
of the stars as a whole. The more intensely luminous
stars are seen and recorded in numbers out of all propor-
tion to their actual abundance in space. It is of great
importance to bear in mind this limitation of statistical
work on star- catalogues ; we ought to consider whether
the results derived from the very special kind of stars that
appear in them may legitimately be extended to the stars
as a whole.

In the same way, the Table gives a very different idea
of the proportions in which the different types of spectra
occur from the impression we should gather by examining
the catalogues. Four stars of Type M are included,
although in the catalogues this class forms only about a
fifteenth of the whole number. On the other hand, Type
B stars (the Orion type), which are rather more numerous
than Type M in the catalogues, have not a single repre-
sentative here. The explanation is that these M stars are
usually very feebly luminous objects (as may be seen in
the Table), and can rarely be seen except in our immediate
neighbourhood. Type B stars on the other hand are
intensely bright, and although they occur but sparsely in
space, we can record even those near the limits of the
stellar system, making a disproportionately large number.
Again in the catalogues the Sirian Type stars (A) and the
Solar Type (F, G, K) are about equally numerous, but in
this definite volume of space the latter outnumber the
former by ten to two.

In order to obtain more extensive data as to the true
proportions of the spectral types, and the relation of
spectral type to luminosity, Table 4 has been drawn up,
containing the stars with fairly well-determined parallaxes

Ill

THE NEAREST STARS

47

between 0"'19 and 0"T1. These are not generally so trust-
worthy as the parallaxes of Table 3, because chief atten-
tion has naturally been lavished on those stars known to
be nearest to us ; but the standard is fairly high, a great
many measured parallaxes being rejected as too uncertain.
Those marked with an asterisk are the best determined
and may be considered equal in accuracy to the parallaxes
of Table 3 ; but as the parallaxes are smaller, the propor-
tionate uncertainty of the calculated luminosity is greater.

TABLE 4.
Stars distant between 5 and 10 parsecs from the Sun.

Star. ^ludT

Spectrum.

Annual
Proper
Motion.

Parallax.

Luminos-
ity
(Sun = l).

f Toucanse . . . 4 "3

F8

2-07

0-15

1-3

*,* Hydri . ... 2'9

G

2-24

0-14

5-4

54 Piscium ... O'l

K

0-59

0-15

0-26

Mayer 20 ... 5 '8

K

1-34

0-16

0-28

*p. Cassiopeire ... 5 '3

G5

375

0-11

1-0

8 Trianguli ... 5'1

G

1-16

0-12

1-0

Pi. 2* 123 ... 5-9

G5

2-31

0-14

0-33

e Eridani ... 4 '3

G5

3-15

0-16

1-15

*5 Eridani ... 3'3

K

075

0-19

2-1

o 2 Eridani .... 4 '5

G5

4-08

0-17

0-84

X Aurig&e .... 4'8

G

0-85

0-11

1-5

*Weisse 5 h 592 . . 8 '9

Ma

2-23

0-18

0-013

Pi. 5 h 146 .... 6-4

G2

0-55

0-11

0-32

*Fed. 1457-8 ... 7 "9

Ma

1*69

0-16

0-042

*Groombridge 1618. 6 '8

K

1-45

0-18

0-09

43 Coma* .... 4'3

G

1-18

0-12

2-2

*Lalande 25372 . . 87

K

2-33

0-18

0-017

Lalande 26196 . . 7'6

G5

0-68

0-14

0-074

*Pi. 14 h 212 ... 5-8

K

2-07

0-17

0-26

*Gronin^en VII.,

No. 20 .... 107

1-22

0-13

0-005

fHerculis .... 3'0

G

0-61

0-14

5-0

*Weisse 17 h 322 . . 7 '8

Ma

1-36

0-12

0-08

70 Ophiuchi ... 4'3

K

1-15

0-17

1-1

*17 Lyrae C. ... 11 '3

175

0-13

0-003

Fomalhaut .... 1*3

A3

0-37

0-14

25-0

*Bradley 3077 - . . 5-6

K

2-11

0-14

0-45

*Lalande 46650 . . 8'9

Ma

1-40

0-18

0-013

Table 4 is far from being a complete list of the stars
within the limits. There should be about 200 stars in

48 STELLAR MOVEMENTS CHAP.

this volume of space, but only 27 are given here.
However, the incompleteness will not much affect the
present inquiry, except that a number of the faintest
stars, which are especially of Types K and Ma, will
naturally be lost. This is noticeable when the list is
compared with Table 3.

Collecting the stars of different spectral types, we have
the following distribution of luminosities from Tables 3
and 4 :

Luminosities.

Type of Parallaxes Parallaxes

Spectrum. greater than 0'20". from 0'19" to O'll".

A ... 48-0

A3 . . 25-0

Ao . . 12-3

F . . . 0-004

F5 . . 9-7

F8 . . 1-4 1-3

G . . . 2-0 *5-4, 5-0, 2-2, 1-5, I'O

G2 . . 0-32

G5 . . 1-15, *1-0, 0-84, 0-33, 0'074

K /0-79 0-5 0-5 0-OOfi I * 2 ' 1 r1 ' *' 45 ' ' 28 ' *' 26

. . O < J, 5, o, . *. *.

K5 . . 0-6, 0-25

Ma . . 0-019, 0-011 0-010, 0*009 *0'08, *0'042, *0'013, *0'013

This summary shows a remarkable tendency towards
equality of brightness among stars of the same type, and
there is a striking progressive diminution of brightness
with advance in the stage of evolution. The single star
of Type F (strictly FO) makes a curious exception. This
star, OA (N) 17415, was measured with the heliometer
by Kriiger as early as 1863 ; apparently the determination
was an excellent one. It would perhaps be desirable to
check his result by observations according to more modern
methods ; but we are inclined to believe that the exception
is real.

It would be tempting to conclude that the great range
in absolute luminosity of the stars is mainly due to
differences of type, and thnt within the same spectral

parallaxes.

in THE NEAREST STARS 49

the range is very limited. We might apply this in
searching for large parallaxes ; for it would seem that,
since stars of Type Ma have so far been found to be of
very feeble luminosity, any star of that class which
appears bright must be very close to us. This hope is
not fulfilled. The bright Ma stars which have been
measured Betelgeuse, Antares, 77 Geminorum and 8 Vir-
ginis have all very small parallaxes, and are certainly much
more luminous even than Sirius, the brightest star in the
Tables. It is perhaps unfortunate that these brilliant
exceptions force themselves on our notice, whilst the far
greater number of normal members of the class are too
faint to attract attention ; we must not be led into an
exaggerated idea of the number of these luminous third
type stars. But it is clear that, notwithstanding the
tendency to equality, if a large enough sample is taken,
the range of luminosity is very great indeed. We shall
have to return to this subject in Chapter VIII.

Table 5 contains some details as to the motions of the
nineteen nearest stars. The transverse velocities are formed
by using the measured parallaxes to convert proper motions
into linear measure.* The radial motions from spectroscopic
observations are added when available. These motions are
relative to the Sun ; if we wished to refer them to the
centroid of the stars, we should have to apply the solar
motion of 20 km. per sec., which might either increase or
decrease the velocities according to circumstances, but
would usually somewhat decrease them. After allowing
for this, the commonness of large velocities is still a strik-
ing and most surprising feature. The ordinary studies of
stellar motions do not lead us to expect anything of the
kind ; and in fact it is not easy to reconcile the general

* The formula, which is often useful, is

Linear speed = annual pjgpermotion_ x 4 . ;4 km per sec
parallax

E

STELLAR MOVEMENTS

CHAP.

investigations with the results of this special study of a
small collection of stars. Taking the fastest moving stars,
Type M, Campbell found an average radial motion of 17
km. per sec. Assuming a Maxwellian distribution of
velocities, this would give for a transverse motion (i.e.,
motion in two dimensions)

Speed greater than

60 km. per sec.
80 ,,

100 ,,

1 star in 53
1 star in 1,100
1 star in 60,000

The presence of 3 stars in the list with transverse
velocities of more than 1 00 km. per sec. (and therefore
certainly more than 80 km. per sec., when the solar motion
is removed) is wholly at variance with the above statistical
scheme.

TABLE 5.
Motions of the nineteen nearest stars.

Star

Proper 5

lotion.

Radial

Arc.

Linear.

Velocity.

Groombridge 34 ....
r\ Cassiopeiae

2-85
1-25

km. per sec.
48
30

km. per sec.
+ 10

I.
I.

T Ceti .

1-93

28

-16

II.

f Eridani

1-00

15

+ 16

II.

CZ 5 h 243 ..

870

129

+ 242

II.

Sirius

1-32

16

7

II

Prooyon
Lalande 21185
Lalande 21258
OA(N) 11677
a Centauri

1-25
477
4-46
3-03
3-66

19
57
106
72
23

-3
-22

1.1

II.
I.
I.
I.

OA (N) 17415 ....
Pos. Med. 2164 . . .
<r Draconis

1-31

2-28
1-84

23
37
43

+ 25

II.
I.
II.

a Aquilae

0-65

T3

-33

I.

61 Cygni
( Indi . . . . .

5-25
4'67

80
79

-62
-39

I.
I

Kriiger 60
Lacaille 9352

0-92
7-02

17
115

+ 12

II.
I.

We cannot attribute the result to errors in the paral-
laxes, for if it should happen that the parallax has

in THE NEAREST STARS 51

been overestimated, the speed will have been under-
estimated ; and it is scarcely likely that any of these
stars have parallaxes appreciably greater than those
assigned in the Table. A possible criticism is that these
stars have been specially selected for parallax-measurement,
because they were known to have large proper motions ;
but the objection has not much weight unless it is seriously
suggested that there are, in this small volume, the hundreds
or even thousands of stars of small linear motion that the
statistical scheme seems to require. Moreover we have
already shown that on the ordinary view as to stellar
motions, seven additional stars would supply the loss due
to the neglect of stars with motions less than 1" per
annum.

Nor can we help matters by throwing over the Max-
wellian law. Originally used as a pure assumption, this
law has been confirmed in the main by recent study of the
radial motions. We should, however, be prepared to admit
that it may not give quite sufficient very large motions.
It has long been known that certain stars, as Arcturus and
Groombridge 1830, had excessive speeds that seemed to
stand outside the ordinary laws. But we find that the
average transverse motion of the nineteen stars is fifty km.
per sec. ; this considerably exceeds the average speed we

should deduce from the stars that come into the ordinary

/

investigations. In round numbers, a mean speed of thirty
km. per sec. relative to the Sun would have been
expected.*

Thus once more it is found that the survey of stars
in the limited volume of space very near to the Sun leads
to results differing from those derived from the stars of
the catalogues. We may fall back on the same explana-

* An average radial speed of 17 km. per sec. gives an average transverse
speed of 26 '5 km. per sec., the factor being ^ whatever the law (Maxwel-

lian or otherwise) of stellar motions. This does not include the solar motion,
which, however, would not increase the result very greatly.

E 2

52 STELLAR MOVEMENTS CHAP.

tion as before, that the catalogues give a very untypical
selection of the stars. But this time the result is more
surprising ; it would scarcely have been expected that
the catalogue-selection, which is purely by brightness,
would have so large an effect on the motions. Yet this
seems to be the case. We notice that the three stars
with transverse speeds of more than 100 km. per sec. have
luminosities 0'007, O'Oll and 0'019. These would be
far too faint to come into the ordinary statistical
investigations. The five stars brighter than the Sun have
all very moderate speeds. Setting the nine most lum-
inous stars against the ten feeblest, we have

Luminosity. Mean transverse speed.

9 Brightest stars . . 48 '0 to 0'25 29 km. per sec.

10 Faintest . . O'lO 0'004 68

It is the stars with luminosity less than l/10th that
of the Sun, with which we are scarcely ever concerned
in the ordinary researches on stellar motions, that are
wholly responsible for the anomaly. The nine bright
stars simply confirm our general estimate of thirty km.
per sec. for the average speed.

The stars of Table 4 also add a little evidence pointing
in the same direction. The parallaxes are scarcely accurate
enough (proportionately to their size) to be used for this
purpose ; but we give the result for what it is worth. It
may be noted that the parallaxes are likely to be a little
overestimated and therefore both luminosities and speeds
will be underestimated. On the other hand, the influence
of selection (on account of large proper motion) will be
greater than in Table 3, tending to increase the mean
speed unduly. There are nine stars given in Table 4 as
having luminosities less than O'l ; their mean speed is
forty-eight km. per sec. Thus these stars have speeds con-
siderably in excess of the thirty km. per sec. originally
expected. They do not, however, include any excessive
speeds.

in THE NEAREST STARS 53

It would be very desirable to have more evidence on
this point before drawing a general conclusion ; but our
task is to sum up the present state of our knowledge,
however fragmentary. The stars, of which the proper
motions and radial velocities are ordinarily discussed, are
almost exclusively those at least as bright as the Sun. It is
generally tacitly assumed that the motions of the far
more numerous stars of less brilliance will be similar to
them. But the present discussion affords a strong sus-
picion that there exists a class of stars, comprising the
majority of those having a luminosity below OT, the speeds
of which are on the average twice as great as the fastest
class ordinarily considered. Apparently the progressive
increase of velocity with spectral type does not end with
the seventeen km. per sec. of the brilliant members of
Type M, but continues for fainter stars up to at least twice
that speed.

In the final column of Table 5. each star has been
assigned to its respective star-stream according to the
direction in which it is moving. We see that eleven stars
probably belong to Stream I, and eight probably to
Stream II. This is in excellent accordance with the ratio
3 : 2 derived from the discussion of the 6000 stars of
Boss's Catalogue. We are not able to detect any significant
difference between the luminosities, spectra, or speeds of
the stars constituting the two streams. The thorough
inter-penetration of the t\vo star-streams is well illustrated,
since we find even in this small volume of space that
members of both streams are mingled together in just
about the average proportion.

REFERENCES. CHAPTER III.

1. Kapteyn and Weersma, Groningen Publications, No. 24.

2. Dyson, Proc. Roy. Sac. Edtnluruh, Vol. 29, p. 378.
:5. Campbell, Stellar Motions, p. 245.

4. Frost, Axh-nit/iit.-ii'rtd J<i,-,i<i.l, Vol. 29, p. 237.

5. Hertzsprung, Attrophyrical Journal, Vol. 30, p. 139.

CHAPTER IV

MOVING CLUSTERS

THE investigation of stellar motions has revealed a
number of groups of stars in which the individual
members have equal and parallel velocities. The stars
which form these associations are not exceptionally near to
one another, and indeed it often happens that other stars,
not belonging to the group, are actually interspersed
between them. We may perhaps arrive at a better
understanding of these systems by recalling a few
elementary considerations regarding double stars.

In only a small proportion of the double stars classed
as " physically connected " pairs, has the orbital motion of
one component round the other been detected. In most
cases the connection is inferred from the fact that the
two stars are moving across the sky with the same proper
motion in the same direction. The argument is that,
apart from exceptional coincidences, the equality of angular
motion signifies both an equality of distance and an
equality of linear velocity. Accordingly the two stars
must be close together in space, and their motions are such
that they must have remained close together for a long
period. Having established the fact that they are
permanent neighbours, we may rightly deduce that their
mutual gravitation will involve some orbital motion, though
it may be too slow to detect ; but that is a subsidiary

CH. iv MOVING CLUSTERS 55

matter, and, in speaking of physical connection, we are not
thinking of two stars tied together by an attractive force.
The connection, if we try to interpret it, appears to be
one of origin. The components have originated in the same
part of space, probably from a single star or nebula ; they
started with the same motion, and have shared all the
accidents of the journey together. If the path of one is
being slowly deflected by the resultant pull of the stellar
system, the path of the other is being deflected at the
same rate, so that equality of motion is preserved. It is
true that the mutual attraction in these widely separated
binaries may help to prevent the stars separating ; but it
is a very feeble tie, and, in the main, community of motion
persists because there are no forces tending to destroy
it.

From this point of view we may have physically con-
nected pairs separated by much greater and even by
ordinary stellar distances, remembering, however, that the
greater the distance the more likely are they to lose their
common velocity by being exposed to different forces. It
is known that exceedingly wide pairs do exist The case
of A Ophiuchi and Bradley 2179 may be instanced ; these
stars are separated by about 14', but have the same
unusually large motion of l r/< 24 per annum in the same
direction. In general it would be difficult to detect pairs
of this kind ; for unless the motion is in some way
remarkable, an accidental equality of motion must often
be expected, and it would be impossible to distinguish the
true pairs from the spurious. It is only when there is
something unusual in the amount or in the direction of
the motion that there are grounds for believing the equality
is not accidental.

In the Moving Clusters we find a closely similar kind of
physical connection. They are considerable groups of
stars, widely separated in the sky, but betraying their
association by the equality of their motions. The most

56 STELLAR MOVEMENTS CHAP.

thoroughly investigated example of a moving cluster is
the Taurus-stream, which comprises part of the stars of
the Hyades and other neighbouring stars. The existence
of a great number of stars with associated motions in this
region was pointed out by R. A. Proctor ; but the researches
of L. Boss have shown the nature of the connection in
a new light. Thirty-nine stars are recognised as belong-
ing to the group, distributed over an area of the sky about
15 square; there can be no doubt that many additional
fainter stars l in the region also belong to the cluster but,
until better determinations of their motions have been
made, these cannot be picked out with certainty.

The first criterion in such a case as this is that the motions
should all appear to converge towards a single point in the
sky ; in Fig. 2 the arrows indicate the observed motions
of these stars, and the convergence is well shown. From
this it may be deduced that the motions are parallel ; for
lines parallel in space appear, when projected on a sphere,
to converge to a point. It is true that the same appear-
ance would be produced if the motions were all converging
to or diverging from a point, but either supposition is
obviously improbable. Theoretically there may be a slight
degree of divergence, the cluster having been originally
more compact ; but calculation shows that on any
reasonable assumption as to the age of the cluster the
divergence must be quite negligible. As the stars are not
all at the same distance from the Sun, the fact that the
speeds are all equal cannot be demonstrated in the same
exact way ; but, allowing for the foreshortening of the
apparent motion in the front of the cluster as compared
with the rear, the proper motions all agree with one
another very nearly. The divergences are just what
we should expect if the cluster extends towards the Sun
and away from it to the same distance that it extends
laterally.

It is clear that in a cluster of this kind the equality ;md

IV

MOVING CLUSTERS

57

c

J C

si C
sJ C

} c
a *

i

3 T

-c

J -c

3

"u

3

\

"T
CJ

\

ID
^

\

I

\

\

\\

I

f

ID

V

s

\

\

5

-\M

\
. \

T

*!

*

\ V

CM
CM

i

.
1

\

|

I

CO

\\

\

T

oc

'

\

\ \

\

TT
U3

\

\

\

I

i
1

in
>

\

\

1

CV

\

t

!

\

\

1

\

C'J

oc

\

\

i

CJ

ur

\

1
1

4 4 \
\ \

1

CO

-r

V x \

rr
CM

\ \

;

in

1

>

58 STELLAR MOVEMENTS CHAP.

parallelism of the motions must be extremely accurate,
otherwise the cluster could not have held together.
Suppose that the motion of one member deviated from the
mean by one km. per sec. ; it would draw away from the
rest of the cluster at the rate of one astronomical unit in
4f years. In ten million years it would have receded ten
parsecs (the distance corresponding to a parallax of O^'IO).
We shall see later that the actual dimensions of the cluster
are not so great as this ; the remotest member is about
seven parsecs from the centre. According to present
ideas, ten million years is a short period in the life even of
a planet like the earth ; the age of the Taurus-cluster,
which contains stars of a fairly advanced type of evolution,
must be vastly greater than this. From the fact that it
still remains a compact group we deduce that the
individual velocities must all agree to within a small
fraction of a kilometre per second.

The very close convergence of the directions of motion
of these stars supports this view ; the deviations may all
be attributed to the accidental errors of observation. In
fact the mean deviation (calculated - observed) in position
angle is 1'8, whereas the expected deviation, due to the
probable errors of the observed proper motions, is greater
than this, a paradox which is explained by the fact that
stars for which the accidental error is especially great
would not be picked out as belonging to the group.

To complete our knowledge of this cluster, one other
fact of observation is required ; namely, the motion in the
line of sight of any one of the individual stars. Actually
six have been measured, and the results are in satisfac-
tory accordance. The data are now sufficient to locate
completely not only the cluster but its individual
members and also to determine the linear motion,
which, as has been shown, must be the same for all the
stars to a very close approximation. This can be done as
follows :

IV

MOVING CLUSTERS

59

The position of the -convergent point, shown at the
extreme left of Fig. 2, is found to be

R.A. 6 h 7 m '2 Dec. +6' 56' (1875'0)

with a probable error of 1*5 chiefly in right ascension.
If is the observer (Fig. 3), let
OA be the direction of this con-
vergent point.

Consider one of the stars S of
the cluster. Its motion * in space
ST must be parallel to OA. Re-
solve ST into transverse and radial
components SX and SY. If SY
has been measured by the spec-
troscope, we can at once find ST
for

ST = SY sec TSY O

= SY sec ACS FIG. 3.

and, since A and S are known points on the celestial
sphere, the angle AOS is known.

Since the velocity ST is the same for every star of the
cluster, it is sufficient to determine it from any one
member the radial velocity of which has been measured.
The result is found to be 45*6 km. per sec.

We next find the transverse velocity for each star,
which is equal to

45 '6 sin AOS km. per sec.

And the stars' distances are given by

transverse velocity = distance x observed proper motion

when these are expressed in consistent units.

It will be seen that the distance of every star is found by
this method, not merely those of which the radial velocity
is known. The distance is found with a percentage accuracy

* The motions considered here are all measured relatively to the Sun.

60 STELLAR MOVEMENTS CHAP.

about equal to that of the observed proper motion ; for the
other quantities which enter into the formulae are very
well-determined. As the proper motions of these stars
are large and of fair accuracy, the resulting distances are
among the most exactly known of any in the heavens.
The parallaxes range from 0"*021 to 0"*031, with a mean
of 0"*025. A direct determination by photography, the
result of the cooperation of A. S. Donner, F. Kiistner,
J. C. Kapteyn, and W. de Sitter, 2 has yielded the mean
value 0"'023::0"'0025, which, though presumably less
accurate than the result obtained indirectly, is a
satisfactory confirmation of the legitimacy of Boss's-
argument.

From these researches the Taurus-cluster appears to be
a globular cluster with a slight central condensation ; its.
whole diameter is rather more than ten parsecs. The
question arises whether this system can be regarded as-
similar to the recognised globular clusters revealed by the
telescope. If there were no more members than the thirty-
nine at present known, the closeness of arrangement of the
stars would not be greater than that which we have found in
the immediate neighbourhood of the Sun. But it appears-
that the 39 are all much brighter bodies than the Sun, and
it is not fair to make a comparison with the feebly
luminous stars discussed in the last chapter. According
to a rough calculation the members of the Taurus-cluster
may be classified as follows :

5 stars with luminosity 5 to 10 times that of the Sun

18 10 20

11 20 50

5 50 100

In the vicinity of the Sun we have nothing to compare
with this collection of magnificent orbs. These stars, it
is true, are separated by distances of the usual order of
magnitude ; but their exceptional brilliancy marks out
this portion of space from an ordinary region. Whether

iv MOVING CLUSTERS 61

there are or are not other fainter members accompanying
them, the term cluster is appropriate enough. There can
be 110 doubt that, viewed from a sufficient distance, this
assemblage would have the general appearance of a
globular star-cluster.

The known motion of the Taurus-cluster permits us to
trace its past and future history. It was in perihelion
800,000 years ago ; the distance was then about half what
it is now. Boss has computed that in 65,000,000 years
it will (if the motion is undisturbed) appear as an ordin-
ary globular cluster 20' in diameter, consisting largely of
stars from the ninth to the twelfth magnitude.

It is interesting to note that a cluster of the size of this
Taurus group must contain many interloping stars not
belonging to it. Even if we omit the outlying members,
the system fills a space equal to a sphere of at least 5
parsecs radius. Now such a sphere in the neighbourhood
of the Sun contains about 30 stars. We cannot suppose
that a vacant lane among the stars has been specially left
for the passage of the cluster. Presumably then the stars
that would ordinarily occupy that space are actually there
non-cluster stars interspersed among the actual members
of the moving cluster. It is a significant fact that the
penetration of the cluster by unassociated stars has not
disturbed the parallelism of the motions or dispersed the
members.

The Ursa Major system is another moving cluster of
which detailed knowledge has been ascertained. It has
long been known that five stars of the Plough, viz., @, 7,
S, e and { Ursae Majoris, form a connected system. By the
work of Ejnar Hertzsprung it has been shown that a
number of other stars, scattered over a great part of the
sky, belong to the same association. The most interest-
ing of these scattered members is Sirius ; and for it the
evidence of the association is very strong. Its parallax
and radial velocity are both well -determined, and agree

62 STELLAR MOVEMENTS CHAP.

with the values calculated from the motion of the whole
cluster. The method by which the common velocity is
found, and the individual stars are located in space, is the
same as that employed for the Taurus-cluster. The
velocity is 18'4 km. per sec. towards the convergent point
R.A. 127'8, Dec. 4- 4 0* 2, when measured relatively to the
Sun. When the solar motion is allowed for, the " absolute "
motion is 28*8 km. per sec. towards R.A. 285, Dec. -2.
As this point is only 5 from the galactic plane, the motion
is approximately parallel to the galaxy.

In Table 6 particulars of the individual stars are set
down, including the parallaxes and radial velocities deduced
by Hertzsprung from the known motion of the system.
In most cases the calculated radial velocities have been
confirmed by observation ; 3 but for nearly all the
parallaxes, it has not been possible as yet to test the
values given. It is quite likely that one or more stars
have been wrongly included ; but there can be little doubt
that the majority are genuine members of the group. The
rectangular co-ordinates are given in the usual unit (the
parsec), the Sun being at the origin, Oz directed towards
the convergent point R.A. 127'8, Dec. +40'2, and Ox
towards R.A. 307'8, Dec. + 49'8, so that the plane zOx
contains the Pole. If a model of the system is made from
these data, it is found, as H. H. Turner 4 has shown, that
the cluster is in the form of a disk ; its plane being nearly
perpendicular to the galactic plane. The flatness is very
remarkable, the average deviation of the individual stars
above or below the plane being 2'0 parsecs, a distance
small in comparison with the lateral extent of the cluster,
viz., 30*to 50 parsecs. In the last column are given the
absolute luminosities in terms of the Sun as unit ; it is
interesting to note that the three stars of Type F are the
faintest, with luminosities 10, !>. ;md 7 respectively.

IV

MOVING CLUSTERS

TABLE C>.
The Ursa Major System.

1

Computed.

Star.

Spec--
trum.

"i. v

Rectangular
Co-ordinates.

Lumin-
osity
(8un=l).

Paral-

Radial

lax.

Velocity:

//

km./Bec.

x y z

3 Eridani . . . 2'92

A2 0-034

-7-5

-13-8 -23-3 12-1

96

.d Auriga- . . . 2 '07

Ap 0-024

-iti-u

7-8 -18-8 36-3

410

Shins .... 1*58

A 0-387

-8'5

-2-0 -I'l 1-2

46

37 Tis;.- Maj. . 5-16

F 0-045

-16-6

7'6 5-8 19-9

7

Urs Maj. . 2 '44

A

0-047

-16-1 7'6 6-9 18-7

76

8 Leonis ... 2 '58

A 2

0-084

-14-4 -2-3 7-1 9-3

21

y Urste Ma]. . 2'.~4

A 0-042

-15-0

8-9 10-5 19-3

87

8 Urste Maj. . . 3 '44

A2

0-045

-14-4

9-8 97 17-2

32

Groom. 1930 .

5-87

F 0-028

-13-4

18-6 15-1 257

9

Urs Mai. - 1'68

Ap

0-042

-13-2

11-4 117 16-9

190

78 Ursjt Maj. . 4 '89

F 0-042

-13-0

12-0 11-8 16-8

10

rrsa? Maj. . .

/2-40
\3-96

~v?| ' 043

-12-2

11-9 12-5 15-3

J93
) 22

a Coronse . . .

2-31

A

0-041

-2-2

12-0 20-9 2-9

110

The stars of the Orion type of spectrum present several
examples of moving clusters. In the Pleiades we have an
evident cluster, in the ordinary sense of the term, and, as
might be expected, the motions of the principal stars and
at least fifty fainter stars are equal and parallel.^ The bright
stars of the constellation Orion itself (with the exception
of Betelgeuse, the spectrum of which is not of TypeB) also
appear to form a system of this sort, the evidence in this
case being mainly derived from their radial velocities, since
the transverse motions are all exceedingly small. In
Orion a faint nebulosity forming an extension of the
Great Orion Nebula, has been discovered, which appears to
fill the whole region occupied by the stars ; it probably
consists of the lighter gases and other materials not yet
absorbed by the stars which are developing. The velocity
of the nebula in the line of sight agrees with that of the

* This is the case as regards the proper motions. The radial velocities of
the six brightest stars show some surprising differences (Adams, Astro-
phijsical Journal, Vol. 19, p. 338), but owing to the difficult nature of the
spectra the determinations are not very trustworthy.

STELLAR MOVEMENTS

CHAP.

stars of the constellation. A similar nebulosity is found in
the Pleiades.

In the case of these, the youngest of the stars, the
argument by which we deduced the accurate equality of
motion in the Taurus-cluster scarcely applies; particularly
in Orion, the dimensions of which must be at least a
hundred times greater than those of the Taurus-cluster, it
is just possible that the associated stars may be dispersing
rather rapidly.

4V

I

FIG. 4. Moving Cluster of " Orion " Stars in Perseus.

A group to which the name moving cluster may be
applied more legitimately is to be found in the constellation
Perseus ; it was detected simultaneously by J. C. Kapteyn,
B. Boss, and the writer. If we examine all the stars of
the Orion type (Type B) in the region of the sky between
K. A. 2 h and 6 h and Dec. + 36 and + 70 (about one-thirtieth
of the whole sphere), we shall find that their motions fall
into two groups. In Fig. 4 the motion of each star is

iv MOVING CLUSTERS 65

denoted by a cross, the star having a proper motion which
would carry it from the origin to the cross in a century.
If all the stars were to start from the origin at the same
instant with their actual observed proper motions, then
after the lapse of a century they would be distributed as
shown in the diagram. Only one star has travelled
beyond the limits of the figure and is not shown; with
this exception the figure includes all the Type B stars in
the region for which data are available.

The upper group of crosses, which is close to the origin,
consists of stars with very minute proper motions, all less
than 1"*5 per century, and scarcely exceeding the probable
error of the determinations. These are clearly the very
remote stars, and there is not the slightest evidence that
they are really associated with one another ; they appear
to cling together because the great distance renders their
diverse motions inappreciable. The lower group consists
of seventeen stars sharing very nearly the same motion
both as regards direction and magnitude. They evidently
form a moving cluster similar in character to those we

o

have considered. Their association is further confirmed
by the fact that'they are not scattered over the whole area
investigated, but occupy a limited region of it.

Table 7 shows the stars which constitute this group. It
has been pointed out by T. W. Backhouse 5 that Nos. 742
to 838 form part of a very striking cluster visible to the
naked eye. The stars a Persei and o- Persei, which are
not of the Orion Type, are included in the visual cluster ;
the motion of the latter shows that it has no connection
with this system, but a Persei appears to belong to it and
may therefore be added to the group. The other stars in this
part of the sky have also been examined so far as possible,
but none of them show any evidence of connection with
the moving cluster. All but three of the stars are
arranged in a sort of chain, which may indicate a flat
cluster (on the plan of the Ursa Major system) seen edge-

66

STELLAR MOVEMENTS

CHAP.

ways. It is always likely that some spurious members
may be included through an accidental coincidence of
motion, and it may be suspected that the three outlying
stars are not really associated with the rest ; on the other
hand, they may well be regarded as original members,
which have been more disturbed by extraneous causes
than the others.

Owing to the small proper motion, the convergent point
of this group cannot well be determined. The motion
deviates appreciably from the direction of the solar antapex ;
so that this cluster possesses some velocity of its own
apart from that attributable to the Sun's own motion.

TABLE 7.
Mooing Cluster in Perseus.

Boss's
No.

Name of Star.

Type.

Mag.

R.A.

Dec.

Centen-
nial
Motion.

Direc-
tion.

h. m.

/>

, o

678

Pi. 220 ....

Bo

5-6

2 54

+ 52

4-3

51

740

30Persei . . .

B5

5-5

3 11

4-44

3-8

55

742

29 Persei . . .

B3

5-3

3 12

+ 50

4-5

52

744

31 Persei . . .

B3

5-2

3 12

+ 50

4-2

51

767

Pi. 37 ...

Bo

5-4

3 16

+ 49

3-6

45

780

Brad. 476 ...

B8

5-1

3 21

+ 49

3-2

47

783

Pi. 56

B5

5-8

3 22

+ 50

4*9

37

790

34 Persei . . .

B3

4-8

3 22

+ 49

4-4

-*

57

796

Brad. 480 ...

B8

6-1

3 24

+ 48

4-7

54

817

v/f Persei . . .

B5

4-4

3 29

+ 48

4-3

42

838

8 Persei ....

B5

3-0

3 36

+ 47

4-6

51

898

Pi. 186 ....

B5

5-5

3 49

+ 48

3-9

40

910

t Persei ....

BO

2-9

3 51

+ 40

3-9

49

947

c Persei ....

B3

4-2

4 1

+ 47

4-4

43

1003

d Persei ....

B3

4-9

4 14

+ 46

4-5

55

1253

15 Camelopardi .

B3

6'4

5 11

+ 58

3-5

37

1274

p Aurigae . . .

B3

5'3

5 15

+ 42

4-5

40

772

a Persei ....

F5

17

3 17

+ 50

3-8

55

In the last column the "direction" is the angle between the direction of
motion and the declination circle at 4 hours R.A.

The tracing of these connections between stars widely
separated from one another is an important branch of

iv MOVING CLUSTERS 67

modern stellar investigation ; and, as the proper motions
of more stars become determined, it is likely that further
interesting discoveries will be made. It seems worth while
at this stage to consider what are the exact criteria by
which we may determine whether a group of stars possesses
that close mutual relation which is denoted by the term
" moving cluster." Since some thousands of proper
motions are available, it must be possible, if we take
almost any star, to select a number of others the
motions of which agree with its motion approximately.
This is especially the case if, the parallax and radial
velocity being unknown, the direction of motion is alone
considered ; but, even if the velocities were known in all
three co-ordinates, we could pick out groups which would
agree approximately ; j ust as in a small volume of gas
there must be many molecules having approximately
identical velocities. Clearly the agreement of the motions
is no proof of association, unless there is some further
condition which indicates that the coincidence is in some
way remarkable. There is a further difficulty that, as we
have already seen in Chapter II. , stars, scattered through
the whole region of the universe that has been studied,
show common tendencies of motion, so that they have
been divided into the two great star-streams. We must
be careful not to mistake an agreement of motion arising
from this general cosmical condition for the much more
intimate association which is seen in the Taurus and Ursa
Major systems.

In the case of the Taurus and Perseus clusters the discrimi-
nation is comparatively simple. These are compact groups
of stars, so that only a small region of the sky and a small
volume of space are considered, and the extraneous stars
which might yield chance coincidences are not numerous.
In the Taurus-cluster the large amount of the motion
makes the group remarkable ; and, although in the
Perseus-cluster the proper motion is not so great, we have

F 2

68 STELLAR MOVEMENTS CHAP.

been careful to show by the diagram that it is very dis-
tinctive. In the latter cluster, moreover, the resemblance
of its members in type of spectrum helped to render the
detection possible.

The Ursa Major system, which is spread over a large
part of the sky, presents greater difficulties. The dis-
crimination of its more scattered members was only
possible owing to the fact that its motion is in a very
unusual direction. Its convergent point is a long way
from the apex of either star-stream and from the solar
apex, and stars moving in or near that direction are rare.
In the writer's investigation of the two star-streams based
on Boss's Preliminary General Catalogue, a striking
peculiarity was presented in one region, which on enquiry
proved to be due to five stars of this system ; that five
stars moving in this way should attract attention,
sufficiently illustrates the fact that motion in this par-
ticular direction is exceptional. We are thus to a large
extent safeguarded from chance coincidences ; never-
theless, our ground is none too certain, and it may
reasonably be suspected that one or two of the members
at present assigned to the group will prove to be spurious.

When the supposed cluster is not confined to one part
of the sky and to one particular distance from the Sun,
when there is nothing remarkable in its assigned motion,
and when the choice of stars is not sufficiently limited, by
the consideration of a particular spectral type or otherwise,
not much weight can be attached to an approximate
agreement of motion. A careful statistical study of groups
in these adverse circumstances may eventually lead to
important results ; but for the present we cannot be satis-
fied to admit clusters the credentials of which do not
reach the standard that has been laid down.

In concluding this chapter we may try to sum up the
importance of the discovery of moving clusters in stellar
a-tromony. An immediate result is that in the Taurus

iv MOVING CLUSTERS 69

and Ursa Hajor stream we have been able to arrive at
precise knowledge of the distance, relative distribution,
and luminosity of stars which are far too remote for the
ordinary methods of measurement to be successful. An im-
portant extension of this knowledge may be expected when
the proper motions of fainter stars have been accurately
determined. Further, the possibility of stars widely
sundered in space preserving, through their whole life-time
up to now, motions which are equal and parallel to an
astonishingly close approximation, is a fact which must be
reckoned with when we come to consider the origin and
vicissitudes of stellar motions. Generally the stars which
show these associations are of early types of spectrum ;
but in the Taurus cluster there are many members as far
advanced in evolution as our Sun, some even of type K,
whilst in the more widely diffused Ursa Major system
there are three stars of type F. Some of these systems
would thus appear to have existed for a time comparable
with the life-time of an average star. They are wandering
through a part of space in which are scattered stars not
belonging to their system interlopers penetrating right
among the cluster stars. Nevertheless, the equality of motion
has not been seriously disturbed. It is scarcely possible
to avoid the conclusion that the chance attractions of stars
passing in the vicinity have no appreciable effect on stellar
motions ; and that if the motions change in course of time
(as it appears they must do) this change is due, not to the
passage of individual stars, but to the central attraction
of the whole stellar universe, which is sensibly constant
over the volume of space occupied by a moving cluster.

REFERENCES. CHAPTER IV.

1. 'rYo/i//<</<'/i Publication*, No. 14, p. 87.

-. (rfaninfien P>il>lii-<ifii>n*, No. 23.

3. Plummer, Monthly X f '., Vol. 73. p. 4<>r>, Table X. (last two columns).

4. Turner, The Observatory, Vol. 34, p. -J4r,.

5. Backhouse, Monthly y<>(i<-<>.-<. Vol. 71, p. 523.

70 STELLAR MOVEMENTS CH. iv

BIBLIOGRAPHY.

Taurus Cluster L. Boss, Astron. Joum., No. 604.

Ursa Major Cluster .... Ludendorff, A sir. Nar,h., No. 4313-14;

Hertzsprung, Astrophytu'id Joiim., Vol.

30, p. 135 (Erratum, p. 320).
Perseus Cluster Kapteyn, Internat. Solar Union, Vol. 3,

p. 215 ; Benjamin Boss, Astron. Journ.,

No. 620 ; Eddington, Monthly Notices,

Vol. 71, p. 43.

There are in addition two less defined clusters, which have attracted some
attention, viz., a large group of Type B stars in Scorpius and Centaurus, and
the 61 Cygni group, which consists of stars scattered over most of the sky
having a linear motion of the large amount 80 km. per sec. 5

Scorpius-Centaurus Cluster . Kapteyn, loc. cit., p. 215; Eddington, loc.

cit., p. 39.
61 Cygni Cluster B. Boss, Astwn. Joum., Nos. 629, 633.

CHAPTER V

THE SOLAR MOTION

IT was early recognised that the observed motions of the
stars were changes of position relative to the Sun, and that
part of the observed displacements might be attributed to
the Sun itself being in motion. The question " What is
the motion of the Sun ? " raises at once the philosophical
difficulty that all motion is necessarily relative. In reality
the manner in which the observed motion is to be divided
between the Sun and the star is indeterminate ; these
bodies are moving in a space absolutely devoid of fixed
reference marks, and the choice and definition of a
framework of reference that shall be considered at rest is
a matter of convention. Probably philosophers of the
last century believed that the undisturbed aether provided
a standard of rest which might suitably be called absolute ;
even if at the time it could not be apprehended in practice,
it was an ultimate ideal which could be used to give
theoretical precision to their statements and arguments.
But according to modern views of the sether this is no
longer allowable. Even if we do not go so far as to discard
the aether-medium altogether, it is generally considered
that no meaning can be attached to the idea of measuring
motion relative to it ; it cannot be used even theoretically
as a standard of rest.

In practice the standard of rest has been the " mean of

71

72 STELLAR MOVEMENTS CHAP.

the stars," a conception which may be difficult to define
rigorously, but of which the general meaning is sufficiently
obvious. Comparing the stars to a flock of birds, we can
distinguish between the general motion of the flock and
the motions of particular individuals. The convention is
that the flock of stars as a whole is to be considered at
rest. It is not necessary now to consider the reasons that
may have suggested that the mean of the stars was an
absolute standard of rest ; it is sufficient to regard it as a
conventional standard, which has considerable usefulness.
If there is any real unity in the stellar system, we may
expect to obtain a simpler and clearer view of the pheno-
mena by referring them to the centroid of the whole rather
than to an arbitrary star like the Sun. By the centroid is
meant in practice the centre of mass (or rather the centre
of mean position) of those stars which occur in the
catalogues of proper motions that are being discussed.
As it is only the motion of this point that is being
considered, its actual situation in space is not of conse-
quence. If the motion of the centroid varied considerably
according to the magnitude of the stars used or the
particular region of the sky covered by the catalogue, it
would be a very inconvenient standard. It is not yet
certain what may be the extent of the variations arising
from a particular selection of stars ; but, as the data of
observation have improved, the wide variations shown in
the earlier investigations have been much reduced or
satisfactorily explained. At the present day, whilst few
would assert that the " mean of the stars " is at all a
precise standard, the indeterminateness does not seem
sufficiently serious to cause much inconvenience.

The determination of the motion of the stars in the
mean relative to the Sun, and the determination of the
solar motion (relative to the mean of the stars) are two
aspects of the same problem. The relative motion, which-
ever way it is regarded, is shown in our observations by a

v THE SOLAR MOTION 73

strong tendency of the stars to move towards a point in
the sky, which according to the best determinations is near
ft Columbae. Although individual stars may move in
widely divergent or even opposite directions, the tendency
is so marked that the mean of a very few stars is generally
sufficient to exhibit it. Sir William Herschel's l first
determination in 1783 was made from seven stars only,
vt't lie was able to indicate a direction which was a good
first approximation. From his time up till recent years
the determination of the solar motion was the principal
problem in all statistical investigations of the series of
proper motions which were measured from time to time.
This investigation was usually associated with a determina-
tion of the constant of precession a fundamental quantity
which is closely bound up with the solar motion in the
analysis. In fact, both quantities are required to define
our framework of reference ; the solar motion defines what
is to be regarded as a fixed position, and the precession-
constant defines fixed directions among the continually
shifting stars. The numerous older determinations of the
solar motion are now practically superseded by two results
published in 1910-11, which rest on the best material yet
available.

The determination by Lewis Boss 2 from the proper
motions of his Preliminary General Catalogue of 6188
Stars gives,

fR.A. 270-5 1-5
SolarA P ex \Dec. +34-3 1-3

The determination by W. W. Campbell 3 from the radial
velocities (measured spectroscopically) of 1193 stars
gives,

fR.A. 268-5 2-0
^olarApex \Dec. +25-3 1-8

Speed of the solar motion 19'50'6 kilometres per second.

The probable errors are not given by Campbell ; but the

74 STELLAR MOVEMENTS CHAP.

foregoing approximate values are easily deduced from the
data in his paper.

The discordance in declination between these two
results, derived respectively from the transverse and the
radial motions, is considerably greater than can be attributed
to the accidental errors of the determinations. Possibly
the discordance may be attributed to the different classes
of stars used in the two investigations. Campbell's result
depends almost wholly on stars brighter than 5 m *0, whereas
Boss included all stars to the sixth magnitude and many
fainter stars. Moreover, Boss's result depends more especially
on the stars nearest to the Sun ; for in forming the mean
proper motion in any region the near stars (having the
largest angular motions) have most effect, whereas in
forming the mean radial motion the stars contribute equally
irrespective of distance. So far as can be judged, however,
these differences will not explain the discordance. Boss
made an additional determination of the solar apex,
rejecting stars fainter than 6 m> ; the resulting position
RA. 269'9, Dec. + 34'6 is almost identical with his
main result. The writer, 4 examining the same proper
motions on the two star-drift theory, by a method which
gives equal weight to the near and distant stars, arrived at
the position R.A. 267'3,' Dec. + 36'4, again a scarcely
appreciable change. The cause of the difference between
the results from the proper motions and the radial motions
thus remains obscure.

One of the most satisfactory features of Boss's deter-
mination of the solar apex is the accordance shown by the
stars of different galactic latitudes. If there is any
relative motion between stars in different parts of the sky,
it would be expected to appear in a division according to
galactic latitude. The following comparison of the results
derived from regions of high and low galactic latitudes is
given by Boss.

v THE SOLAR MOTION 75

Solar Apex.

Galactic Latitude of Zones. R.A. Dec.

- 7 D to +7 D 269 40' +33 17'

-19 to - 7 3 , and +19 to + 7 270 55 29 52

-42 to -19, and +42 to +19 269 51 34 18

S. Gal. Pole to -42, and N. Gal. Pole to +42 270 32 36 27

The differences are quite as small as could be expected
from the accidental errors.

Another comparison of different areas of the sky can
be made from the results of an analysis by the two-drift
theory, which has the advantage that the position depends
equally on all the stars used, instead of (as in the ordinary
method) the nearest stars having a preponderating share.

We find,-

Solar Apex.
Region. R.A. Dec.

P*rA g*t5Sig} 265-5 + 37-0

Equatorial Area Dec. -36 to + 36 3 269 '4 + 36'4

There is thus a very satisfactory stability in the position
of the apex determined from different parts of the sky.*
The evidence is less certain as to its dependence on the
magnitude and spectral type of the stars. There is some
indication that the declination of the apex tends to increase
for the fainter stars ; but it is not entirely conclusive.
The range of magnitude in Boss's catalogue is scarcely grea't
enough to provide much information ; so far as it goes, it
is opposed to the view that there is any alteration in the
apex for stars of different magnitudes, for, as already
mentioned, the stars brighter than 6 m '0 give a result
almost identical with that derived from the whole
catalogue. From the Groombridge stars (Dec. +38 to
N. Pole), F. \V. Dyson and W. G. Thackeray 5 found-

* In the foregoing comparisons antipodal regions have always been taken
together. There remains a possibility of a discordance between opposite
hemispheres.

76 STELLAR MOVEMENTS CHAP.

Solar Apex.

Magnitude. , s

m. in. R.A. Dec. No. of Stars.

1-0-4-9 245 3 +16-0 200

5-0-5-9 268 + 27 = '0 454

6-06-9 278 + 33-0 1,003

7-0-7-9 280 +38^5 1,239

8-08-9 272 3 +43-0 811

This shows a steady increase in declination with diminishing'
magnitude. It must, however, be noted that the area covered
by the Groombridge Catalogue is particularly unfavourable
for a determination of the declination of the apex.

Other evidence pointing in the same direction has been
found by G. C. Comstock, who made a determination of
the solar motion from 149 stars of the ninth to twelfth
magnitudes. He was able to obtain proper motions of
these stars, because they had been measured micro-
metrically as the fainter companions of double stars, but
had been found to have no physical connection with the
principal stars. The resulting position of the apex is

R.A. 300 Dec. +54

In a more recent investigation 7 the same writer has used
479 faint stars, with the results

Magnitude 7 m> to 10 m '0 Apex R.A. 280 Dec. +58

10 m -0 13 m -0 R.A. 288 : Dec. +71

The weight of these determinations cannot be great, but
they tend to confirm the increase of declination with the
faintness of the stars.

Earlier investigations in which the stars were classified
by magnitude are those of Stumpe and Newcomb. Tint
former, using stars of large proper motion only, found a
considerable progression in declination with faintness.
Newcomb, on the other hand, who used stars of small
proper motion only, found that the declination is steady.
We now know that, owing to the phenomenon of star-
streaming, the exclusion of stars above or below certain
limits of motion is not legitimate, so that the contradictory
character of these two results is not surprising.

v THE SOLAR MOTION 77

The comparative uncertainty of the proper motions of
the fainter stars requires that results based on them
should be received with caution. In particular, since the
mean distance of the stars increases with faintness, the
average parallactic motion becomes smaller, and a sys-
tematic error in the declinations of any zone has a greater
effect on its apparent direction. This is particularly
serious, because these investigations have been usually
based on northern stars only or on an even less extensive

region.

R.A.

Dec.

No. of Stars.

274= -4

+ 34 -9

490

270-0

28-3

1,647

265 3 '9

28-7

656

259-3

42-3

444

275-4

40-3

1,227

273-6

38-8

222

There is fairly consistent evidence that the declination
of the solar apex depends to some extent on the spectral
type of the stars, being more northerly for the later types.
In Boss's investigation 8 the following results were found,

Solar Apex.

Type.

Oe5 B5

B8 A4

A5-F9

G

K

M

The later types G, K, M thus yield a declination differing
markedly from the earlier types ; or, if we prefer to set
aside the results for the groups containing few stars, which
may be subject to large accidental errors, and confine
attention to types A (B8 A4) and K, the difference of
12 between the results of these two classes is evidently
significant.

The results of Dyson and Thackeray from the Groom-
bridge stars show the same kind of progression.

Solar Apex.

Type. R'A. ~~De7. No. of Stars.

B, A 269 +23^ 1100

F, G, K 27M :;; 866

Other investigations of this relation depend mainly on
stars now included in Boss's catalogue, and used in his

7 8 STELLAR MOVEMENTS CHAP.

discussion. It therefore does not seem necessary to quote
them.

To sum up the results we have arrived at, it appears
that we can assign a point in the sky at about R.A. 270,
Dec. +34 towards which the motion of the Sun relative to
the stellar system is directed. For some reason at present
unknown, the determinations of this point by means of the
spectroscopic radial velocities differ appreciably from those
based on the transverse motions, giving a declination
nearly 10 lower than the point mentioned. When
different parts of the sky are examined the results are
generally in good agreement, so that there can be little
relative motion of the stars as a whole in different regions.
There is some evidence that the solar apex increases in
declination as successively fainter stars are considered, and
it seems certain that for the later types of spectrum the
declination is higher than for the earlier types. From all
causes the solar apex from a special group of stars may
(apart from accidental error) range from about +25 to
+ 40 in declination ; variations in right ascension appear
to be small and accidental.

The speed with which the Sun moves in the direction
thus found can only be measured from the radial motions.
The result derived from the greatest amount of data is
19 '5 km. per sec.

Attention has been lavished on the investigation of the
solar motion, not only on account of its intrinsic interest,
but also because it is a unit of much importance in many
investigations of the distribution of the more distant stars.
The annual or centennial motion of the Sun is a natural
unit of comparison in dealing with the stellar system,
generally superseding the radius of the earth's orbit, which
is too small to be employed except for a few of the nearest
stars. It provides a far longer base-line than can be
obtained in parallax-observations ; for the annual motion
of the Sun amounts to four times the radius of the earth's

v THE SOLAR MOTION 79

orbit, and the motion of fifty or a hundred years, or even
longer, may be used. The apparent displacement of the
star attributable to the solar motion is called the
parallactic motion. By determining the parallactic
motion (in arc) of any class of stars their average
distance can be found, just as the distance of an indi-
vidual star is found from its annual parallax. It is not
possible to find by observation the parallactic motion of an
individual star, because it is combined with the star's
individual motion ; but for a group of stars which has no
systematic motion relative to the other stars, these indi-
vidual motions will cancel in the mean.

It may be appropriate to add some remarks on the
theory of the determination of the solar motion from the
observations. The method usually adopted for discussing
a series of proper motions is that known as Airy's.

Take rectangular axes, Ox being directed to the vernal
equinox, Oy to R.A. 90, and Oz to the north pole. The
parallactic motion (opposite to the solar motion) may be
represented by a vector, with components X, Y, Z, directed
to the solar antapex. X, Y, Z are supposed to be expressed
in arc, so as to give the parallactic motion of a star at a
distance corresponding to the mean parallax of all the stars
considered.

Taking a small area of the sky let the mean proper
motion of the stars in the area be /* a , ^ in right ascension
and declination respectively. Then considering the pro-
jection of (X, F, Z) on the area considered, we have

- X sin a + Y cos a = /u a

X cos a sin 8 Y sin a sin d + Z cos 6 = pi

where it is assumed that these stars are at the same mean
distance as the rest, and that their individual motions
cancel out. If these assumptions are not exactly satisfied,
the deviations are likely to be mainly of an accidental

:

8o STELLAR MOVEMENTS CHAP.

character. Taking the above equations for each region, a
least-squares solution may then be made to determine
X, Y, Z. The right ascension and declination A, D of the
solar antapex are given by

tan A = Y/X
tanD = Z;(X*+Y*)l

Additional terms in the equations of condition involving
the correction to the preoessional constant and to the
motion of the equinox are often inserted, but these need
not concern us here. When stars distributed uniformly
over the whole sky are considered, the additional terms have
no effect on the result.

Although the argument is clearer when we use the
mean proper motion over an area for forming equations of
condition, it is quite legitimate to use each star separately.
For it is easily seen that the resulting normal equations
are practically identical in the two procedures. In using
the mean proper motion, it is easier and more natural to
give equal weights to equal areas of the sky instead of
weighting according to the number of stars ; this is
generally an advantage. Further the numerical work is
shortened.

There are two weak points in Airy's method. First the
mean proper motion (which, if not formed separately for
each area, is virtually formed in the least-squares solution)
is generally made up of a few large motions and a great
number of extremely small ones. It is therefore a very
fluctuating quantity, the presence or omission of one or
two of the largest motions making a big difference in the
mean. In a determination based nominally on 6000 stars,
the majority may play only a passive part in the result,
and the accuracy of the result is scarcely proportionate to
the great amount of material used. The second point is
more serious, since it leads to systematic error. We have
, i umed that the mean parallax of the stars in each area
differs only by accidental fluctuations from the mean of

v THE SOLAR MOTION 81

the whole sky ; but this is not the case. The stars near the
galactic plane have a systematically smaller parallax than
those near the galactic poles.

It has often been recognised that this property of the
galactic plane may cause a systematic error in the apex
derived from the discussion of a limited part of the sky.
Perhaps it is not so generally known that it will also cause
error even when the whole sky is used. It seems worth
while to examine this point at length. Happily it turns
out that the error is not very large, but this could scarcely
have been foreseen.

If the mean parallax in any area is p times the average
parallax for the whole sky, we may take account of the
variation with galactic latitude by setting

p = l + f P 2 (cos 6)

where 6 is the distance from the galactic pole, e is a
coefficient and

We shall consider the case when the observations extend
uniformly over the whole sky.

The equations of condition should then read

Xp sin a + Yp cos a = fi a

Xp cos a sin 8 Yp sin a sin 8 + Zp cos 8 = /ig.

We wish to re- interpret the results of an investigator who
has not taken p into account. We therefore form normal
equations, just as he would do, viz., from the right
ascensions :

X'S.p sin -a - Y^p sin a cos a = - S^i a sin a

X"S,p sin a cos a + YZp cos -a = S/j a cos a.

and from the declinations :

X^p cos -'a sin -8 4- YZp sin a cos a sin -8 Z'S.p cos a sin 5 cos 8 =

- 2^5 cos a sin 8
X2p sin a cos a sin 2 8 + Y^.p sin 2 a sin -8 Z2p sin a sin 8 cos 8=

2^s sin a sin 8

- A'2p cos a sin 8 cos 8 - Y2p sin a sin 8 cos 8 + ZXp cos -8 = i>6 cos 8

G

82 STELLAR MOVEMENTS CHAP.

giving the combined equations :

X'Zp (sin 2 a + cos -a sin 2 8) Y2p sin a cos a cos 2 8 Z'S.p cos a sin 8 cos 8 =

- 2(/u a sin a + ^5 cos a sin 8)

X"2.p sin a cos a cos 2 8 + Y2p (cos 2 a + sin -a cos 2 8) Z'S.p sin a sin 8 cos 8 =

2(/i a cos a ps sin a sin 8)
- X'S.p cos a sin 8 cos 8 - YZp sin a sin 8 cos 8 + Z^.p cos -8 = 2/xg cos 8.

Now it clearly can make no difference in a least-squares
solution whether we resolve our proper motions in right
ascension and declination or in galactic latitude and
longitude. The value of the solar motion, which makes
the sum of the squares of the residuals in R.A. and Dec.
a minimum, must be the same as that which makes the
sum of the squares of the residuals in Gal. Lat. and Long.
a minimum. We may therefore treat a solution as though
it had been made in galactic co-ordinates, although the
actual work was done in equatorial co-ordinates.

Let then a, S now stand for galactic longitude and
latitude, so that A", F, Z is the paral lactic motion vector
referred to rectangular galactic co-ordinates. We shall
have

Taken over a whole sphere the mean value of

2 1

p(sin -a + cos 2 a sin 2 8) = - + ~e

3 15

3 lo

cos 2 8 = - -_? e

3 15

The other coefficients vanish when integrated over a
sphere. Thus the normal equations become (setting N for
the total number of stars used)

2 / 1 \

x JT ( 1 + J~Q J = - 2 (pa sin a + (i& cos a sin 8)-r- N

2 r / 1 \

o Y ( 1 -f jTj J = 2 (/* a cos a ps sin u sin 8) -r- N

v THE SOLAR MOTION 83

And if X , y o , ZQ are the solutions obtained when the P 2
term is neglected,

The original and corrected galactic latitudes of the antapex
bein \ , X, we have

tan X =

l-t<

whilst the galactic longitude is unaltered.

The effect of the decrease of parallax towards the galactic
plane is thus to make X numerically less than \. The
uncorrected position of the solar apex is too near the
galactic plane.

Inserting numerical values, X = 20, and e may perhaps
be ^ (i.e., mean parallax at the pole / mean parallax in the
plane =8/5), we find X = 2157 r . The correction is
just under 2. Reverting to equatorial co-ordinates, the
correction is mainly in right ascension, the right ascension
given by the ordinary solution being about 2'4 too great.

It is quite practicable to work out the corresponding
corrections, when the proper motions cover only a zone of
the sky limited by declination circles. In this case we
have to retain equatorial co-ordinates throughout, and
express P 2 (cos 6) in terms of a and S. The mean values
of the functions of sin a, cos a, sin S and cos S that occur
are readily evaluated for the portion of the sphere used.
As the numerical work depends on. the particular zone
chosen, we shall not pursue this matter further.

A second method of finding the solar apex from the
proper motions, known as Besse.l's method, has been used
by H. Kobold. 9 Each star is observed to be moving along
a great circle on the celestial sphere. Consider the poles
of these great circles. If the stellar motions all converged
to a point on the sphere, the poles would all lie along the

G 2

84 STELLAR MOVEMENTS CHAP.

great circle equatorial to that point. Thus a tendency of
the stars to move towards the solar antapex should be
indicated by a crowding of the poles towards the great
circle equatorial to the antapex. This affords a means of
finding the direction of the solar motion by determining
the plane of greatest concentration of the poles. . It is to
be noted, however, that this method makes no discrimina-
tion between the two ways in which a star may move along
its great circle. Two stars moving in exactly opposite
directions will have the same pole. A paradoxer might
argue that, as the effect of the solar motion is to cause a
minimum number of stars to move towards the solar apex,
the solar motion will be indicated by a tendency of the
poles to avoid the plane equatorial to its direction. How-
ever, if the individual motions are distributed according to
the law of errors, the crowding to the plane will be found to
outweigh the avoidance, so that the method is legitimate

o o

though perhaps a little insensitive. But if the individual
motions follow some other law, the result may be altogether
incorrect. In the light of modern knowledge of the
presence of two star-streams, Bessel's method can no
longer be regarded as an admissible way of finding the
solar apex ; but it is interesting historically, for in
Kobold's hands it first foreshadowed the existence of the
peculiar distribution of stellar motions which is the
subject of the next chapter.

The determination of the solar motion from the radial
velocities presents no difficulty. If (X, Y, Z] is the vector
representing the parallactic motion in linear measure, each
Mar yields an equation of condition :

X cos a cos d + Fsinacos5-H^sin^ = radial velocity.

A least-squares solution is then made, the individual motions
of the star> IM-MI- treated as though they were accidental
errors. The numerical work can be shortened by using in
the equations of Condition tin- nu-an radial velocity for a

v THE SOLAR MOTION 85

sniiill area of the sky, instead of the individual results.
The resulting normal equations are practically unaltered,
and there is no theoretical advantage.

REFERENCES. CHAPTER V.

1. Sir W. Herschel, Collected Papers, Vol. 1, p. 108.

2. Boss, Ash-on. Journ., Nos. 612, 614.

3. Campbell, Lick Bulletin, No. 196.

4. Ecldingtoii, Monthly AW/<v.s, Vol. 71, p. 4.

5. Dyson and Thackeray, Monthly Notices, Vol. 65, p. 428.

6. Comstock, Astron. Journ., No. 591.

7. Comstock, Astron. Journ., No. 655.

8. Boss, Astron. Journ., Nos. 623-4.

9. Kobold, Nova Ada der Kais. Leop. Carol. Deutschen Akad., Vol. 64 ;
Atr. Nach., Nos. 3163, 3435, 3591.

BIBLIOGRAPHY.

The following references are additional to those quoted in the Chapter.
Owing to the recent improvement in the data of observation, and the change
of theoretical views due to the recognition of star-streaming, the interest
of these papers is now, perhaps, mainly historical.

Argelander, Memoires presenter a VAcad. des. Sci., Paris, Vol. 3, p. 590

(1837).

Bravais, Liouvilles's Journal, Vol. 8 (1843).
Airy, Memoirs R.A.S., Vol. 28, p. 143 (1859).
Stumpe, Astr. Nach., No. 3000.
Porter, Cincinatti Trans., No. 12.
L. Struve, Memoires St. Petersboury, Vol. 35, No. 3 ; Astr. Xach. Nos. 3729,

3816.

Newcomb, Astron. Papers of the American Ephemeri*. Vol. 8, Pt. 1.
Kapteyn, Astr. Nach., Nos. 3721, 3800, 3859.
Boss, Astron. Journal, No. 501.
Weersma. Qroningen Publications, No. 21.

CHAPTER VI

THE TWO STAR STREAMS

THE observed motion of any star can be regarded as
compounded of two parts ; one part, which is attributable
to the motion of the Sun as point of- reference, is the
parallactic motion ; and the other residual part is the
star's motus peculiaris or individual motion. It must
be borne in mind that this division cannot generally be
effected in practice for the proper motion of a star ;
because, although the parallactic motion in linear measure
is known, we cannot tell how much it will amount to in
angular measure, unless we know the star's distance,
and this is very rarely the case. On the other hand, the
spectroscopic radial velocities, being in linear measure, can
always be freed from the parallactic motion, if desired.
As the greater part of our knowledge of stellar movements
is derived from the proper motions, we cannot study the
mottis peculiares directly, but must deduce the phenomena
respecting them from a statistical study of the whole
motions.

In researches on the solar motion, it has usually, though
not always, been assumed that the mottis peculiares of the
stars are at random. This was the natural hypothesis
to make, when nothing was known as to the distribution
of these residual motions ; and certainly, when we consider

N

CH. vi THE TWO STAR STREAMS 87

how vast are the spaces which isolate one star from
its neighbour, and how feeble must be any gravitational
forces exerted across such distances, it might well seem
improbable that any general tendency or relation could
connect the individual motion of one star with another.
Yet many years ago the phenomenon of local drifts of stars,
or, as they are now called, Moving Clusters, was known.
But although such instances of departure from the strict
law of random distribution of motions must have been
recognised as occurring exceptionally, probably few
astronomers doubted that the hypothesis was substantially
correct. In 1904, however, Prof. J. C. Kapteyn l showed
that there is a fundamental peculiarity in the stellar
motions, and that they are not even approximately
haphazard. This deviation is not confined to certain
localities, but prevails throughout the heavens, wherever
statistics of motions are available to test it.

Instead of moving indiscriminately in all directions, as
a random distribution implies, the stars tend to move in
two favoured directions. It does not matter whether the
parallactic motion is eliminated or not. A tendency to
move in one favoured direction would disappear, when
the parallactic motion was removed ; but a tendency in
two directions can only be an intrinsic property of the
individual motions of the stars. It may seem strange
that this striking phenomenon was so long overlooked by
those who were working on proper motions ; but usually
investigators, having the solar motion mainly in their
minds, as a first step towards gathering their data into
a manageable form grouped together the stars in small
regions of the sky, and used the mean motion. This
unfortunately tends to conceal any peculiarity in the
individual motions. In order to exhibit the phenomenon
it is necessary to find some means of showing the statistics
of the separate stellar motions ; this may be done conveni-
ently in the following way.

88

STELLAR MOVEMENTS

CHAP.

ANALYSIS OF PROPER MOTIONS

Confining our attention to a limited area of the sky, so
that the apparent motions are seen projected on what is
practically a plane, we count up the number of stars
observed to be moving in the different directions. If in
classifying directions we proceed by steps of 10, we shall
then form a table of the number of stars moving in 36
directions, towards position angles 0, 10, 20 . . . 350.

Velocity 0-3 Unit

Velocity 0-6 Unit

Velocity 1-5 Unit

FIG. 5. Simple Drift Curves.

The result can be conveniently shown on a polar diagram,
i.e., a curve is drawn so that the radius is proportional to
the number of stars moving in the corresponding direction.
Before considering the diagrams actually derived from
observation, let us examine what form of curve would be
obtained if the hypothesis of random motions were correct.
The curve would not be a circle because of the parallactic
motion ; for, as these observed motions are referred to the
Sun, there would IM superposed on the random individual
motions the motion of the star-swarm as a whole. If, for
example, this latter motion were towards the north, then

vi THE TWO STAR STREAMS 89

clearly there would be a maximum number of stars moving
north and fewest south, the number falling off symmetri-
cally on either side from north to south. The exact form
can be calculated on the hypothesis of random distribution ;
it varies with the magnitude of the parallactic motion
compared with the average motus peculiaris, being more
elongated the greater the parallactic motion. In Fig. 5

270
240 300

ISO-

ISO 30

12tf I 60

90

FIG. 6. Observed Distribution of Proper Motions.
(Groombridge Catalogue II. A. 14 h to 18 h , Dec. +38' to +70'.)

examples of this curve are given ; it may be noted how
sensitive is the form of the curve to a small change in the
parallactic velocity. It is convenient to have a name for a
system such as is represented in these figures, in which
the individual motions are haphazard, but the system as a
whole is in motion relative to the Sun ; we call such a
system a drift.

As an example of a curve representing the observed
distribution of proper motions we take Fig. 6. 2 This

9 o STELLAR MOVEMENTS CHAP.

corresponds to a region of the sky between K.A. 14 h and
18 h , Dec. 4- 38 and 4- 70, the proper motions being taken
from Dyson and Thackeray's " New Reduction of Groom-
bridge's Catalogue." The motions of 425 stars are here
summarised. It is quite clear that none of the single
drift curves of Fig. 5 can be made to fit this curve
derived from observation. Its form (having regard to
the position of the origin) is altogether different. No one
of the theoretical curves corresponds to it in even the
roughest manner. It will be noticed that there are two
favoured directions of motion ; the stars are streaming in
directions 80aud 225, the latter being the more pronounced
elongation. The actual number of stars moving in each
direction is given in the fifth column of Table 8 below.
Neither of the favoured directions coincides with that
towards the solar antapex, viz., 205, for this part of the
sky. It is true the mean motion of all these stars is
towards the antapex, but we see that that is merely a
mathematical average between the two partially opposed
streams that are revealed in the diagram.

It is possible to obtain a theoretical figure that will
correspond approximately with Fig. 6 in the following
way. Suppose, instead of the single drift we have
hitherto considered, there are two star-drifts. Let one,
consisting of 202 stars, be moving in the direction 225
with velocity* 1*20, and the other, consisting of 232 stars,
be moving in the direction 80 with the much smaller
velocity 0*45. The corresponding curves are P and Q,
Fig. 7. If these were seen mingled together in the sky,
the resulting distribution would be represented by the
curve R. Each radius of R is, of course, formed by
adding together the corresponding radii of P and Q. If
R is carefully compared with the observed curve, it will
be seen that the resemblance is close. The numerical

* The unit velocity l/7i, which is related to the mean individual motion, is
denned in the mathematical theory in the next chapter.

VI

THE TWO STAR STREAMS

comparisons, which these diagrams illustrate, are given in
Table 8 ; it is there shown that by adding together two

TABLE 8.

Analysis of Proper Mat inns in the Region R.A. 14* to 18 A ,
Dec. +38 to +70.

Calculated.

Direction.

Observed.

Difference.
Obs. Calc.

Drift I.

Drift II.

Total.

5

066

4

-2

15

077

5 -2

25

088

6

-2

35

10 10

9 -1

45

11 11

10 -1

55

12 12

14 +2

65

12 12

14 +2

75

13 13

14 +1

85

13 13

13

95

12 12 12

105

1 12 13

10 -3

115

1 11 12

11 -1

125

1

10 11

10 -1

135

1

8

9

10 +1

145

2

7

9

7 -2

155

3

6

990

165

5

6

11

9 -2

175

7

5

12

14 +2

185

11

4

15 14 - 1

195

15

4

19 16 -3

205

19

3

22 21

-1

215

23

3

26 27 +1

225

24

3

27 29

+ 2

235

23

3

26 26

245

19

3

22 19

-3

255

15

3

18 17

-1

265

11 3

14

12

-2

275

7 3

10 11

+ 1

285

5 3 8 11

+ 3

295

3 3

6 8

+ 2

305

2 3

5 7

+ 2

315

1346

+ 2

325

1 4

5 6

+ 1

335

1 4

5 5

345

1

5

6 5

-1

355

6

6

4

-2

Totals .

. 202

232

434 425

92 STELLAR MOVEMENTS CHAP.

theoretical drifts, the observed distribution of motions is
approximately obtained. Without pressing the conclusion
that a combination of two simple star-drifts will represent
the actual distribution in all its detail, we may at least
assert that it represents its main features, whereas not
even the roughest approximation can be obtained with

Q

FIG. 7. Calculated Distributions of Proper Motions.

a single drift, i.e., with the hypothesis of random-
motions.

As another illustration, we may take Fig. 8, which refers
to a different part of the sky, the proper motions this time
being taken from Boss's Preliminary General Catalogue.
The uppermost curve, which has so interesting an appear-
ance, is derived from the observed proper motions. Curve
B is the best approximation that can be found on the
assumption of a random distribution of motions plus
the parallactic motion. It may be remarked that since
the solar apex is a rather well determined point, the
direction of elongation of the curve B is not arbitrary ; it
is necessary to draw it pointing in the known direction of
the parallactic motion. The curve C is an approximation
by a combination of two star-drifts ; these again were not
taken as arbitrary in direction, but were made to point
towards appropriate apices deduced from a general discus-
sion of the whole sky. It is quite probaldr that there are
differences between A and C which are not purely

VI

THE TWO STAR STREAMS

93

accidental, but it will
at least be admitted
that, whereas the curve
B bears scarcely any
resemblance to the ob-
served curve, the curve
C - reproduces all the
main features of the
distribution, and from
it we can if we choose
proceed to investigate
the irregularities of de-
tail.

The foregoing examples
illustrate a method of
analysis which has been
applied successfully to a
great many parts of the
sky. It consists in find-
ing, generally by trial
and error, a combination
of two drifts which will
give a distribution of
motions agreeing closely

with that actually observed. In comparing the results
obtained from different parts of the sky, it must
be remembered that we are studying the two-
dimensional projections of a three-dimensional phenom-
enon, and the diagrams will vary in appearance as
the circumstances of projection vary. The most accu-
rate series of proper motions at present available
is contained in Lewis Boss's " Preliminary General
Catalogue," and it is of special interest to examine fully
the results derivable from it. 3 The catalogue contains
6188 stars well-distributed over the whole sky; practically

Double-drift approximation

FIG. 8. Observed and Calculated
Distributions of Proper Motions.
(Boss, Region VIII.)

94 STELLAR MOVEMENTS CHAP.

all stars down to the sixth magnitude are included, and the
fainter stars appear to be fairly representative and have
not been selected on account of the size of their proper
motions. A very high standard has been attained in the
elimination of systematic error the main cause of trouble
in these researches though no doubt there is still a
possibility of improvement in this respect. There can be
no doubt that the catalogue represents the best data that
it is at present possible to use.

After excluding certain classes* of stars for various
reasons, 5322 remained for investigation. These were
divided between seventeen regions of the sky, each region
consisting of a compact patch in the northern hemis-
phere together with an antipodal patch in the southern
hemisphere. By taking opposite areas together in this
way we double the number of stars without unduly
extending the region, for the circumstances of projection
in opposite regions are identical. The regions were
numbered from I. to XV IT., I. being the circular area
Dec. 4- 70 to the Pole; II. to VII. formed the belt
between + 36 = and +70 with centres at O 1 ', 4 h , 8 h , 12 h ,
16 h , and 20 h respectively; and VIII. to XVII. formed the
belt to +36 with centres at l h 12 ra , 3 h 36'", etc.
(These are the positions of the northern portions ; the
antipodal part of the sky is also to be included in each
case.)

The diagrams for 11 of the 17 regions are given in
Fig. 9. The arrows marked Antapex point to the antapex
of the solar motion (R.A. 90*5, Dec. 34 *3) ; the arrows
I. and II. point to the apices of the two drifts, found from
the collected results of this discussion. It will be seen that
the evidence for the existence of two star-streams is very
strong. The tendency to move in the directions of the
arrows I. and II. is plainly visible, and it is scarcely

* Viz., stars of the Orion type, members of moving clusters, and the fainter
components of binary systems.

VI

THE TWO STAR STREAMS

95

necessary again to emphasise that there is no resemblance
to a symmetrical single-drift curve pointing along the
antapex arrow. In certain cases, notably Regions XIV.

270"

i. Region I. ; centre, North Pole.
270

180

ii. Region II. ; centre, R.A. O h , Dec. +50 3 .

270

180

II

in. Region V. ; centre, R.A. 12 h , Dec. + 50.

FIG. 9. Diagrams for the Proper Motions of Boss's " Preliminary
General Catalogue."

STELLAR MOVEMENTS

270 270

180

180

iv. Region VI. ; centre R.A. 16 h ,
Dec. +50.

270

\

180 -,

v. Region VII. ; centre R.A. 20 1 ',
Dec. +50.

II

A

Antsipex
vi. Region VIII. ; centre, R.A. l h 12 m , Dec. +17.

270

180

II

90

vii. Region XII. ; centre, R.A. 10 h 48 m , Dec. +17 .

Fit;. 9 (continued). Diagrams for the Proper Motions of Boss'
"Preliminary

THE TWO STAR STREAMS

II

viii. Region XIII. ; centre R.A. 13 h 12 m , Dec. +17.

270

180

90 II

ix. Region XIV. ; centre R.A. 15 h 36 m , Dec. +17.

180

>-

x. Region XVI. ; centre R.A. 20 h 24 m , Dec. +17

FIG. 9 (continued) . Diagrams for the Proper Motions of Boss's
"Preliminary General Catalogue."

H

9 8

STELLAR MOVEMENTS

270

CHAP,

180

xi. Region XVII. ; centre R.A. 22 h 48 m , Dec. +17.

FIG. 9 (continued). Diagrams for the Proper Motions of Boss's
" Preliminary General Catalogue/'

and XVI. , there appears to be a streaming towards the
antapex in addition to the streaming in the directions of
the two drifts, so that the -curve appears three-lobed like
a clover leaf. This is an important qualification of our
conclusion, but for the present we shall not discuss it ;
later it will be considered fully. The eleven regions
chosen for representation are those in which the separ-
ation into two drifts ought to be most plainly indicated.
It will be understood that there are parts of the sky in
which the projection is such that they are not very plainly
separated. In fact there must be one plane of projection
on which the drifts have identical transverse motions, and
would therefore become indistinguishable, except by
having recourse to the radial velocities. The fact, then,
that in the regions which are not here represented the
phenomenon is shown less plainly, in no way weakens the
argument, but rather confirms it.

Let us suppose now that w r e have succeeded in analysing
the stellar motions in each of the seventeen regions into
their constituent drifts, and have thus determined the
directions and velocities of the two drifts at seventeen
points of the celestial sphere. If the drift motion in each
region is really the same motion seen in varying projec-
tions, we must find that <>n plotting the directions on a

VI

THE TWO STAR STREAMS

99

globe they will all converge to one point. This will be true
for each drift separately. The convergence actually found is
shown in Figs. 10 and 11. Imagine the great circles traced

+ 30

+20-

+ 10-

o 5 10 15 20'

Approximate Scale

FIG. 10. Convergence of the Directions of Drift I. from the 17 Regions.

+65

+50

+55

+45

5 1 15

Approximate Scale

FIG. 11. Convergence of the Directions of Drift II. from the 17 Regions.

on the sky and a photograph taken of the part of the sky
where they converge ; the great circles on such a photograph
will appear as straight lines. These are shown on the

H 2

ioo STELLAR MOVEMENTS CHAP.

two diagrams, and the Roman numeral attached to each
line indicates the region from which it comes. Each
diagram represents an area of the sky measuring about 60
by 30 ; this would correspond on the terrestrial globe to
a map of Northern Africa from the Congo to the Mediter-
ranean. The apex marked on each diagram is the defini-
tive apex of the drift, determined by a mathematical
solution. For one of the drifts, called Drift L, the con-
vergence of the directions is so evident as to need no
comment. Owing to the smaller velocity of Drift II., its
direction in any region cannot be determined so accu-
rately, and a greater deviation of the great circles must be
expected. Having regard to this, the agreement must be
considered good, Region VII. being the only one showing
important discordance. To appreciate the evidence of
this diagram we may make a terrestrial comparison ; if
from seventeen points distributed uniformly over the
earth tracks (great circles) were drawn, every one of which
passed across the Sahara, they might fairly be considered
to show strong evidence of convergence ; the distribu-
tion of the Drift II. directions is quite analogous.

The analysis of the regions gives not only the directions
of the two drifts, but also their speeds in terms of a
certain unit ; and both sets of results may be used in
finding definitive positions of the apices towards which
the two drifts are moving. The results of a solution by
least squares are as follows :

Drift I. Drift II.

Apex.

Apex.

R.A.

Dec.

Speed.

R.A

Dec.

Speed.

10 Equatorial Regions

( .>2

4

-14

1

1

507

286

5

-63

6

0-869

7 Polar Regions . .

HU-

3

- 1<>

7

1

536

289

1

83

r>

O'16

Whole Sphere . .

GO'

8

-14 C

6

1

516

287

8

-64

1*

1 0-855

* The fact that the declination derived from tho whole sphere does not lie
between the declinations from the two portions looks paradoxical, but is due
to the unequal weights of the determinations of the Z component from the
two portions.

vi THE TWO STAR STREAMS >oi

The speeds are measured in terms of the usual theo-
retical unit 1 //.

The drift-speed in an} 7 region should (owing to fore-
shortening) vary as the sine of the angular distance from
the drift-apex, being greatest 90 away from the apex
and diminishing to zero at the apex and antapex. This
progressive decrease as the regions get nearer the apex is
well shown in the observed values, and the sine-law is
followed with very fair accuracy.

Another fact derived from the analysis is the proportion
in which the stars are divided between the two streams ;
this seems to vary somewhat from region to region, but
the mean result is that 59*6 per cent, belong to Drift
I. and 40*4 per cent, to Drift II. ; that is a proportion of
practically 3:2.

To sum up, the result of this analysis of the Boss
proper motions is to show that the motions can be closely
represented if there are two drifts. That which we have
called Drift I. moves with a speed of 1'52 units, the other,
Drift II., with a speed of 0'86 unit. The first drift
contains f of the stars and the second drift f . Their direc-
tions are inclined at an angle of 100.

It will be remembered that these motions are measured
relative to the Sun. In Fig. 12, let SA and SB repre-

B c A

FIG. 12.

sent the drift-velocities, making an angle of 100. Divide
AB at C so that A C : CB = 2:3 corresponding to the
proportion of stars in the two drifts. Then SO represents

STELLAR MOVEMENTS CHAP.

the motion of the centroid of all the stars relative to the
Sun, and accordingly CS represents the solar motion
and points towards the solar apex. AB and BA repre-
sent the motion of one drift relative to the other ; the
points in the sky towards which this line is directed are
called the Vertices. The positions found from the num-
bers above are

v fR.A. 94>-2 Dec.

3 ' ' \R.A. 274=-2 Dec. -11 '9.

The relative velocity of the two drifts is 1*87 units.

It is a remarkable fact that the vertices fall exactly
in the galactic plane, so that the relative motion of the
two drifts is exactly parallel to the galactic plane.

The solar motion CS found from the same numbers is
0*91 towards the

Solar Apex. . . R.A. 267 C '3 Dec. +36 '4

This may be compared with Prof. Boss's determination
from the same catalogue by the ordinary method 4 :

Solar Apex. . . R.A. 270'5 Dec. + 34-3

The agreement is interesting because the principles of
the two determinations are very different ; moreover, in
Boss's result the magnitudes of the proper motions as
well as their directions were used, whereas the analysis
on the two-drift theory depends solely on the directions.

Since the speed of the solar motion has also been
measured in kilometres per second, this provides an
equation for converting our theoretical unit into linear
measure. We have 0*91 unit =19*5 km./sec., whence
the theoretical unit l/h is 21 kilometres per second. We
can thus, if we wish, convert any of the velocities
previously given into kilometres per second.

It will b(- seen from Fig. 12 that the direction of motion
of Drift I., SA, is inclined at a comparatively small angle
to the parallactic motion, SC. But that the directions

vr THE TWO STAR STREAMS 103

are clearly distinct may be appreciated by referring back
to Fig. 10. The solar an tapex actually falls just outside
that diagram, so it is clear that the convergence is not
towards the solar antapex but towards a different apex at
the point indicated.

When referred to the centroid of the stars instead of to
the Sun, the motions CA and CB of the two drifts are
seen to be opposite to one another. It is perhaps not
easy to realise that the inclination of the two stream-
motions is a purely relative phenomenon depending on
the point of reference chosen ; but this is the case. If we
divest our minds of all standards of rest and contemplate
simply two objects in space two star-systems all that can
be said is that they are moving towards or away from or
through one another along a certain line. The distinction
between meeting directly or obliquely disappears. It is
clear that this line joining the vertices must be a very
important and fundamental axis in the distribution of
stellar motions. It is an axis of symmetry, along which
there is a strong tendency for the stars to stream in one
direction or the other. It is this point of view T that has
led to an alternative mode of representing the phenomenon
of star-streaming, the ellipsoidal theory of K. Schwarz-
schild. 5

We have, hitherto, analysed the stars into two separate
systems, which move, one in one direction, the other
in the opposite direction, along the line of symmetry ;
but Schwarzschild has pointed out that this separation is
not essential in accounting for the observed motions. It
is sufficient to suppose that there is a greater mobility
of the stars in directions parallel to this axis than in
the perpendicular directions. The distinction is a little
elusive, when it is looked into closely. It may be illus-
trated by an analogy. Consider the ships on a river.
One observer states that there are two systems of ships
moving in opposite directions, namely, those homeward

io 4 STELLAR MOVEMENTS CHAP.

bound and those outward bound ; another observer makes
the non-committal statement that the ships move
generally along the stream (up or down) rather than
across it. This is a not unfair parallel to the points
of view of the two-drift and ellipsoidal hypotheses. The
distinction is a small one and it is found that the two
hypotheses express very nearly the same law of stellar
velocities ; but by the aid of different mathematical
functions. This will be shown more fully in the mathe-
matical discussion in the next chapter. Meanwhile we
may sum up in the words of F. W. Dyson 6 : " The
dual character of Kapteyn's system should not be unduly
emphasised. Division of the stars into two groups was
incidental to the analysis employed, but the essential
result was the increase in the peculiar velocities of stars
towards one special direction and its opposite. It is this
same feature, and not the spheroidal character of the
distribution, which is the essential of Schwarzschild's
representation."

The phenomenon of star-streams (by which we mean the
tendency to stream in two favoured directions, which both
the two-drift and ellipsoidal theories agree in admitting)
is shown very definitely in all the collections of proper
motions that are available for discussion. Very careful
attention has been given to the question whether it could
possibly be spurious, and due to unsuspected systematic
errors in the measured motions. 7 It is not difficult for the
investigate* to satisfy himself that such an explanation is
quite out of the question, but it is not so easy to give the
evidence in a compact form. Happily we are able to
give one piece of evidence which seems absolutely con-
clusive. F. W. Dyson 8 has made an investigation of the
stars (1924 in number) with proper motions exceeding 20"
a century. In this case we are not dealing with small
quantities just perceptible with refined measurements, but

VI

THE TWO STAR STREAMS

105

with large movements easily distinguished and cheeked.
These large motions show the same phenomenon that has
been described for the smaller motions of Boss's catalogue.
In fact, the two streams are shown more prominently when
we leave out the smaller motions. This does not mean
that the more distant stars are less affected by star-
streaming than the near stars ; it is easy to show that for
stars at a constant distance the small proper motions must

N

Scale
20" 30" 40"

FIG. 13. Distribution of Large Proper Motions (Dyson).

necessarily be distributed more uniformly in position angle
than the large motions, and the enhancement of the stream-
ing when the small motions are removed is due to this cause.
The diagram Fig. 13 is taken from Dyson's paper ; it
refers to the area R. A. 10 h to 14 h ,Dec. -30 to +30. The
motion of each star is represented by a dot, the displace-
ment of the dot from the origin representing a century's
motion on the scale indicated. The blank space round the
origin is, of course, due to the omission of all motions less
than 20" ; we can imagine it to be filled with an extremely

io6 STELLAR MOVEMENTS CHAP.

dense distribution of dots. It is clear that the distribution
shown in the diagram represents a double streaming
approximately along the axes towards 6 h and S. No
single stream from the origin could scatter the dots as

o o

they actually are. Although the general displacement is
towards the solar antapex (i.e., towards the lower right
corner), this is accompanied by an extreme elongation of
the distribution in a direction almost at right angles.
Thus the phenomenon of two streams is well shown in the
largest, and proportionately most trustworthy, motions that
are known, so that it requires no particular .delicacy of
observation to detect it.

RADIAL MOTIONS

Doubtless the most satisfactory confirmation of this
phenomenon found in the transverse motions of the
stars would be an independent detection of the same
phenomenon in the spectroscopically measured radial
velocities. Although great progress has been made in the
determination and publication of radial velocities, the
stage has scarcely yet been reached when a satisfactory
discussion of this question is possible. We shall see that
the results at present available are quite in agreement
with the two-stream hypothesis, and afford a valuable
confirmation of it in a general way ; but a larger amount
of data is required before we can see how precise is the
agreement between the two kinds of observations.

In the transverse motions the two streams were detected
by considering the stars in a limited area of the sky ; there
was no need to go outside a single area, except at a later
stage when it was desired to show that the different parts
of the sky were concordant. But with the radial veloci-
ties, we can learn nothing of star-streaming from a single
region ; that is the drawback of a one-dimensional
projection compared with a two-dimensional. To pass
from one area to another involves questions of stellar dis-

VI

THE TWO STAR STREAMS

107

tribution, which complicate the problem. In particular it
is necessary to pay attention to spectral type. It is well
known that the early type stars are more numerous near
the galaxy than elsewhere ; as these have on the average
smaller residual motions than the later types, there will be
a tendency for the radial motions near the galactic plane
to be smaller than near the poles. But evidence of star-
streaming should be looked for in a tendency for the residual
radial velocities to be greater near the vertices (which are
in the galactic plane) than elsewhere. The two effects are
opposed, and there is a danger that they will mask one
another.

By treating the different types separately the difficulty
is avoided, but in that case the data become rather
meagre. For Type A the results have been worked out by
Campbell 9 who gives the following table :

TABLE 9.
A veraye Residual Velocities. Type A .

Stitude Distance from Kapteyn's Vertices.

0-30
3060=
6090

030

30-60

60 -90"

IS^

10-3 33
11'2 4 6

11 -735 km. per sec.
7^
9-3^

Mean . .

IS^

10'8 79

Q.X

y ioo

The suffixes show the number of stars.

The increased velocity in the neighbourhood of the
vertices seems to be plainly marked, and it is in fair
quantitative agreement with the results from the transverse
velocities.* A division of the results according to galactic

* There seems to have been some misunderstanding on this point. It is
considered mathematically in Chapter VII.

108 STELLAR MOVEMENTS CHAP.

latitude is given, showing that the progressive increase is
not dependent upon that.

The radial velocities of the later types of spectrum have
not yet been discussed.

GENERAL CHARACTERISTICS OF THE MEMBERS OF THE
Two STREAMS

We now turn to the question whether there is any
physical difference between the stars of the two streams.
Are they of the same magnitude and spectral type on
the average ? And are they distributed at the same
distance from the Sun, and in the same proportions in
all parts of the sky ? It is not possible on the two-
drift theory definitely to assign each star to its proper
drift ; all that can be said is that, of the stars moving in a
specified direction, a certain proportion belong to Drift I.
and the rest to Drift II. There are, however, directions in
which the separation is nearly complete, and, by confining
ourselves to these, we may pick out a sample of stars
of which 90 per cent, or more belong to Drift L, and
another sample equally representative of Drift II. Our
samples are thus not absolutely pure, but they are
sufficient for testing whether there is any physical differ-
ence between the members of the two drifts.

By thus separating the drifts as far as possible, the
following table has been constructed to compare the magni-
tudes of the stars. The stars are those of Boss's catalogue
(stars of Type B being omitted). Approximately equal
samples of each drift have been taken from each of ten
regions ; the results for the five polar and five equatorial
regions are shown separately in Table 10.

The last two columns of the table agree closely ; the
slight excess of very bright stars in Drift I. may be
noticed, but it seems to be of an accidental character.

VI

THE TWO STAR STREAMS

109

TABLE 10.

MmjnittnJ.es of Stars of the Two Streams.

Polar B

Equatorial Regions.

'? g TvTT VIIL IX - XIL

Total.

I. II. V. \

XIII. & XVII.

Magnitude.

Drift I.

Drift II. Drift I.

Drift II.

Drift I.

Drift II.

0-02-9

16

7

4

22

11

3-03-9

17

10

10

12

27

22

4-04-9

46

52

38

38

84

90

5-05-4

50

52 39

52

89

104

5-45-9

99

100

78

68

177

168

6-06-4

75

72

75

79

150

151

6-56-9

50

59

57

41

107

100

7-0

44

51

52

49

96

100

Variable

7

2

2

7

4

Total .

404

405

355

345

759

750

10

The investigation may be extended to somewhat
fainter stars by taking the Groombridge proper motions.
Samples taken from these give :

Number of stars of magnitudes

03-9. 4-04-9. 5-05-9. 6-06-9. 7'0 7'4. 7'5 7'!'. 8'0 8'4. S'5 8'9. Total.

Drift I. 16 29 86 171 136 108 104 51 701
Drift II. 3 23 81 169 125 113 132 61 707

The excess of very bright stars is again noticed, but
this is to some extent a repetition of the stars which
occurred in the last table and is not fresh evidence.
Further the Type B stars were not excluded in forming
this table ; these form a considerable proportion of the
bright stars, and their motions are known to be peculiar.
At the other end of the table an excess of faint stars
in Drift II. is shown, but it is not very decided. This
may be spurious, being the effect of a greater accidental
error in determining the directions of motion of the
faintest stars, which tends to increase falsely the number
assigned to the slower-moving drift.

no

STELLAR MOVEMENTS

CHAP.

The main conclusion from the two tables is that there
is no important difference in the magnitudes of the
stars constituting the two drifts. On the other hand
there is possibly some significance in the fact that
discussions of the faint stars, such as the Groombridge
and Carrington stars, have given a higher proportion
belonging to Drift II. than the discussions of brighter
stars, such as those of Bradley and Boss.

The same course may be pursued with regard to
spectral types, though, in view of the known differences
in the amount of the individual motions of late and
early type stars, such a treatment is unsatisfactory, and
the interpretation of the result is ambiguous. We give,
however, results for four regions of the Groombridge

stars :

TABLE 11.
Spectra of Stars of the Two Streams.

Drift I.

Drift II.

Region.

No. of

No. of

Percent-

No. of No. of

Percent-

stars.

stars.

age of stars. stars.

age of

Type I. Type II.

Type II. Type I. Type II.

Type II.

A

61

66

52

36 70 .

66

B

95

35

27

61 45

42

C

61

23

27

16 16

50

G

58

39

40

41 39

49

In each case the percentage of Type II. (later type)
stars is higher in Drift II. than in Drift I. But some
caution is needed in interpreting this table. It may
be that the result is due to the elements of the star-
stream motions differing from one type to another.
We cannot tell whether the distribution of spectra
differs from one drift to the other, or whether the
drift-motions differ from one spectral type to another.
This matter is still undecided ; but, knowing at least

vi THE TWO STAR STREAMS in

one remarkable relation between type and motion, we can-
not ignore the second alternative.

A safe way of stating the conclusion is, that stars of
early and late types are found in both streams, but
that there is a somewhat higher proportion of late types
among stars moving in the direction of Drift II. than
of Drift I.

DISTANCES OF THE Two STREAMS

It is most important to determine whether the two
streams are actually intermingled in space. It might, for
example, be suggested that one of the streams consists
of a cluster of stars surrounding the Sun, which moves
relatively to the background of stars constituting the other
stream. The absence of any appreciable correlation between
magnitude and drift renders such an explanation rather
improbable, for it would be expected that the stars of the
background would be fainter on the average than those of
the nearer swarm. The question can, however be treated
more definitely by using the magnitude of the proper
motions to measure the distances of the two drifts. Hitherto
we have only made use of the directions of the motions
without reference to the amount ; we must now bring the
latter element into consideration.

Let G?! and d. 2 be the respective mean distances * of the
two drifts ; if these are known, the theory (set forth in
the next chapter) enables us to calculate the mean
proper motion of stars moving in any direction. Take
for instance Fig. 14, which refers to a region of the
Groombridge catalogue ; t the curves are drawn so
that the radius vector in any direction measures the
mean proper motion in the corresponding direction.
The velocities of the drifts and the numbers of stars

* Unless otherwise specified the mean distance of a system of stars means
the harmonic mean distance, or distance corresponding to the mean parallax.
f This is the "restricted Region G," Monthly Notices, Vol. 67, p. 52.

ii2 STELLAR MOVEMENTS CHAP.

belonging to each have first been found by the usual
method ; we can then draw the theoretical mean proper
motion curves for any assumed values of d l and d. 2 . Two
such curves are shown, viz., for d l = d 2 and for d l = d. 2 .
The first curve A has a slight bi-lobed tendency, that is to
say, there are two directions in which the radius vector is
a maximum ; but it will be seen that the mean proper
motion curve is not a very sensitive indicator of the presence
of two streams. That does not matter for our present
purpose. The upper part of the curve arises mainly from
stars of Stream II., the lower part from Stream I. If we
decrease the average distance of Stream I. and increase

B C

FIG. 14. Mean Proper Motion Curves.

A. Theoretical Distribution. Two Drifts at the same Mean Distance.

B. Theoretical Distribution. Second Drift at twice the Distance of First
Drift.

C. Observed Distribution.

that of Stream II. the lower part of the curve will expand
and the upper part shrink. This is what has happened in
curve B.

The remaining curve represents the results of observa-
tion. We have purposely selected a region containing
a large number of stars (767), so that the observed
curve is fairly smooth ; but a mean proper motion is
nearly always liable to large accidental fluctuations, and
we must not expect a very close agreement with theory, it
will be noticed that C is much more definitely bi-lobed

VI

THE TWO STAR STREAMS

than the theoretical curves; the two streams are more
prominent than was anticipated. This phenomenon is
found not only in this particular example, but in all the
divisions of the Groombridge motions. It suggests that
our two-drift analysis has not succeeded in giving a
complete account of the facts. (The ellipsoidal hypo-
thesis fails equally in this respect.) The precise signifi-
cance of the failure has not been made out ; we cannot do
more than call attention to an outstanding difference.

The difference in shape of the curves A and C renders
a comparison somewhat difficult, but it will be recognised
that the proportion of the two lobes is not very different,
pointing to approximately equal distances for the two
drifts. There is no such magnification of one lobe at the
expense of the other as is illustrated in curve B.

The values of d t and d 2 can be obtained by a rigorous
mathematical solution by least squares. The results for
this region (G) and six others are given in Table 12. 12

TABLE 12.
Mean Parallaxes of the Two Drifts.

Limits of Region. Drift I.

Drift II.

Region.

Dec. N.

P . 1 Probable
MI Error.

1

us.

Probable
Error.

h

*

. "

A

7090

024 2-96 0-07

3-41

0-13

B 3870

22 2 2-45 0-07

2-40

0-11

r^

38-70

2 6 2-39 0-08

2-65

0-12

D 3870

610 3-35 0-12

3-23

0-21

E 3870

1014 3-65 0-14

478

0-29

F 3870

14_18 374 0-23

4-15

0-31

G 3870

18-22 2-77 0-16

2-55

0-18

The quantities \jhd l and l/hd. 2 are the mean parallaxes
multiplied by a factor the value of which is probably about
450. As, however, some large proper motions have been

I

n 4 STELLAR MOVEMENTS CHAP.

excluded (the same proportion from each drift), the absolute
parallaxes have no definite significance. It is the ratios
that are of importance.

The table shows that in each region the two drifts are
at nearly the same mean distance. In only one case,
Region E, is the difference at all significant, and even
there the ratio, about 4:3, necessitates a very considerable
intermixture of the two sets of stars. Moreover, Region
E contains fewer stars than any other, and the results are
in consequence uncertain. It is thus necessary to regard
the two star-streams as thoroughly intermingled systems,
and to throw aside any hypothesis which regards them
as passing one behind the other in the same line of
sight.

The table also shows a variation in mean distance from
region to region, which is greater than the variation from
Drift I. to Drift II. This variation is found to follow the
galactic latitude of the stars, and is due to the fact
that we see a larger number of more distant stars the
nearer we get to the plane of the Milky Way. As both
drifts show this progressive change, these distant stars
must belong to both drifts impartially.

A similar conclusion is derived from the more recent
and accurate proper motions of Boss's catalogue. In
order to obtain a large enough number of stars we have
thrown together Regions VIII., XII., XIII. and XVII. of
the previous division, and considered the large region
consisting of two antipodal areas each about 70 square.
This region is one of high galactic latitude, so that the
proper motions are comparatively large ; and, its centre
being nearly 90 from both Apex and Vertex, the large
extent of the area is less harmful than it would be in
other parts of the sky. This region contains 1122 stars.

In Table 13 the second column gives the number of stars
moving in each 10 sector. The third column gives the mean
proper motion of these stars ; in order to smooth out minor

VI

THE TWO STAR STREAMS

irregularities, these means have been taken for overlapping
30 sectors-. The fourth column gives the calculated
mean proper motion, based on the known velocities and

TABLE 13.

Mean Proper Motions of a Region of Boss's Catalogue
(Centre of Region, R.A. 0\ Dec. 0).

Direction.

1 No. of Stars.

Mean Centennial P.M.

Observed.

Calculated.

c

94

12-73

12-61

10

90

12-20

12-61

20

88

11-15

12-54

30

66

10-27

12-32

40

79

10-56

11-45

50

50

8-92

10-51

60

41

9-49

9-35

70

34

8-82

8-99

80

33

10-00

8-91

90

30

11-19

9-13

100

25

10-85

9-35

110

33

11-24

9-57

120

34

874

9-57

130

36

8-98

9-35

140 50

9-09

9-35

150 25

9-56

8-99

160 32

8-90

8'55

170 13

7-05

8-12

180 10

190 6

200 7

210

4

220

1

230

4

5-98

6-23

240

6

250

9

260

10

270

8

280

6

290

4

6-42

6-09

300

20

7-33

6-88

310

10

7-83

8-05

320

14

9-17

9-20

330

26

10-40

10-15

340

44

11-27 11-16

350 80

12-46

12-03

I 2

ti6 STELLAR MOVEMENTS CHAP.

relative proportions of the drifts, and on the assumption
that they are at the same mean distance. As the number
of stars moving in directions between 175 and 285 is too
small to give trustworthy separate mean proper motions,
these have been combined to give one mean.

The corresponding polar diagrams are given in Fig. 15.
The agreement is very fair ; but, as in the previous case,

Theoretical distribution Observed distribution

FIG. 15. Mean Proper Motion Curve (Region of Boss's Catalogue).

the two drifts appear more sharply in the observed curve
than in the theoretical one. It is clear that our assump-
tion of equal distances for the two drifts cannot be far
wrong. A rigorous solution by least squares leads to the
results :

For Drift I. . JL = 6" '94 0"'10 per century

ha

Drift II i = 7"'38 0"-17

Or, adopting the value of the unit l/h already found,
viz., 21 km. per sec., the mean parallaxes are

Drift I CT0156 0" '00023

Drift II 0"-0166 0'''00038

On the question whether the proportion of mixture
remains the same in all parts of the sky, conflicting
views have been held. We are not concerned with
local irregularities, but with any general tendency of

VI

THE TWO STAR STREAMS

one drift to prevail over a hemisphere or belt of the
sky. It seems to be agreed that there is no systematic
connection between the numbers and galactic latitude.
Table 14 taken from the analysis of Boss's catalogue
shows this clearly. The individual irregularities may

TABLE 14.
Division of the Stars between the Drifts.

Boss's
Region.

No. of Stars.

Ratio
Drift II. : Drift I.

Galactic
Latitude.

III.

304

0-60

o

1

X.

356

0-66

1

VIT.

448

0-87

9

[I.

354

0-48

12

XVI.

311

0-69

14

XV.

285

071

17

I.

371

0-60

27

IX.

275

0-75

29

XI.

365

0-42

31

, IV.

294

0-87

33

XVII.

245

0-77

37

VIII.

308

0-67

44

VI.

294

0-52

46

- XIV.

259

0-68

48

XII.

342

0-64

61

V.

274

1-03

66

XIII.

237

0-59

78

|

be partly real and partly the result of insufficient data
or errors in the proper motions ; but there is no syste-
matic progression. As already explained, these regions
consist each of two antipodal areas, and hence
the results do not allow us to test whether there is a
difference between two opposite hemispheres of the sky.
According to S. S. Hough and J. Halm 13 there is a con-
siderable difference. From a discussion of the radial
velocities of stars they deduced that the Drift If. stars
were concentrated in the hemisphere towards the point R. A.
3:24 , Dec. 12 ; the range of density was not explicitly
determined but it was evidently considerable, the Drift

n8 STELLAR MOVEMENTS CHAP.

II. stars being relatively two or three times as numerous
at one pole compared with the other. This result
depended, at least partly, on an apparent general
dilatation of the stellar system or excess of positive
radial velocities compared with negative. At the present
time the excess is more generally attributed to a
systematic error in the radial velocities of certain types
of stars, possibly attributable to a pressure displacement
of the spectral lines. 'The result cannot, therefore, be
regarded as trustworthy in itself. Analysis of the angular
motions, however, confirmed the general conclusion.

The same authors, 14 from an examination of the
Bradley proper motions, found a maximum density
of Drift II. at approximately O h R.A. and in a southern
declination, possibly the south galactic pole ; they
showed further that this inequality of distribution would
fully account for certain anomalies found by Newcomb
in his discussion of the precessional constant, viz., a
difference in the results from the right ascensions and
declinations, and a residual twelve-hour term 0"'50
cos S cos 2a in the mean proper motions in right
ascension. In his latest researches Halm has adopted
a more complicated three-drift hypothesis ; his results
(based now on the Boss proper motions) still indicate
a very marked excess of Drift II. stars towards about
22 h in general agreement with his former conclusions.

To summarise this discussion of the distribution and
characters of the stars of the two streams, we may
say that on the whole the mixture is remarkably
complete. Such differences as are found are difficult to
interpret with certainty, being liable to be influenced
by small systematic errors in the proper motions ; and
in most cases it is scarcely possible as yet to discriminate
between a difference in the proportion of the drifts ,-md
a difference in their velocities. An excess of later
spectral type stars in Drift II., and n relative excess of

vi THE TWO STAR STREAMS 119

Drift II. stars in the southern galactic hemisphere, are
the most important differences found ; but there are
indications that this crude interpretation of statistical
results is not an adequate description of the complex
distribution of motions in different parts of the stellar
system and among different classes of stars.

The recognition of two star-streams may be expected to
throw some light on the discrepancies in earlier determina-
tions of the solar motion. Of these one of the most
remarkable is that due to H. Kobold, 15 who in fact was led
by it to a partial recognition of the systematic motions of
the stars. Using Bessel's method, Kobold found for the
solar apex the position R.A. 269, Dec. -3 , differing by
at least 35 from the position generally accepted. We
have shown (p. 83) that Bessel's method depends very
essentially on the individual motions of the stars being
distributed nearly in accordance with the law of errors.
As that is now known to be untrue, it is not surprising
that the point found is very different from the real apex.
It is easy to see that, but for the solar motion, Bessel's
method would give a very sensitive determination of the
vertex ; and it is really much better adapted for deter-
mining the vertex than the apex. The point found by
Kobold is indeed quite close to the vertex found on the
two-drift or ellipsoidal hypotheses, the presence of the
solar motion having produced a deviation of only a few
degrees. There is another way of looking at the matter.
In applying BesseFs method to determine the solar motion,
only the line joining the apex and antapex is found, and
there is nothing to indicate which end is the apex. If
there are two drifts, we might expect that the line deter-
mined by Kobold would be a sort of weighted mean
between the corresponding lines of the two drifts. This is
actually the case. But naturally Kobold's line is in the
acute angle between the drift-axes, whereas the solar

120 STELLAR MOVEMENTS CHAP.

motion is the other mean lying in the obtuse angle
between them.

It was customary in early determinations of the solar
apex to consider separately the proper motions between
various limits of size. It was always found that the
declination of the apex decreased as the magnitude of
the proper motions increased. This is easily explained on
the two-drift theory. Since Drift I. has much the greater
velocity relative to the Sun, the stars belonging to it have
on an average greater proper motions than the Drift II.
stars. Thus the larger the motions discussed, the greater
will be the proportion of Drift I. stars, and the nearer will
be the resulting apex to the Drift I. apex.

In like manner the higher declination of the solar apex
of the later type stars may be accounted for by the
increased proportion of Drift II. in the later types. The
higher declination of the apex for faint stars, and the
higher proportion of faint stars in Drift II. (both somewhat
doubtful deductions from the observations) would also
correspond.

THREE-DRIFT HYPOTHESIS.

From what has been said as to the relations of the
two-drift and ellipsoidal hypotheses, it will be understood
that we do not regard the analysis as giving more
than an approximation to the actual law of stellar
motions. Its importance is that it takes account of
what is clearly the most striking feature of the distri-
bution. It has been found, however, that one systematic
deviation can be detected, of which neither of these
hypotheses is able to take account ; we are indeed ready
to take a step forward towards a second approximation.
In comparing the observed distribution with that calcu-
lated on the two-drift or ellipsoidal theories, it is found
that there is always some excess of stars moving towards
the solar antapex. This is shown in the diagrams of the Boss

vr THE TWO STAR STREAMS 121

proper motions (and, in fact, the Groombridge motions)
by a bulge of the curve roughly in the antapex direction.
This was attributed by the writer 16 to a small third stream,
of much less importance than the two great drifts. Halm ir
has shown, however, that it is better to re-analyse the
motions on the assumption of three drifts. According to
his representation, there is a third drift, Drift 0, which is
practically at rest in space, and is thus intermediate
between the two original drifts. Naturally in introducing
the third drift into the analysis some of the stars originally
included in Drift I. and Drift II. are regrouped with Drift
0, and the elements of the two former become somewhat
modified. This is especially so with Drift I., which moves
more nearly in the same direction as Drift 0, and it is
mainly at its expense that the new drift is formed.

That Halm's interpretation of the phenomenon is the
right one maybe seen by looking at the diagrams for Regions
XIV. and XVI. (Fig. 9, ix and x). In most cases Drift I.
and Drift overlap so much that they appear almost as one
drift ; only a little extra bulge of the curve towards the
antapex betrays the duality. Regions XIV. and XVI. are
the two places where their directions become more open,
and in these the three distinct streams of about equal
importance are plainly manifest. Remembering that it is
only in these two regions that we could expect Drift I. and
Drift to be really separate, the evidence for the three-
drift hypothesis becomes very strong. Moreover the third
drift was postulated by Halm, not so much to explain
these peculiarities, but because it seemed needed to
reconcile results derived from different parts of the
celestial sphere.

The hypotheses of two-drifts and of three-drifts, or we
should rather say the two successive approximations, may
be compared thus : In Fig 16, the line CS, which
represents the solar motion, is the same in each case. In
the first diagram we have Stream I. with f of the stars

122

STELLAR MOVEMENTS

CHA

to the Sun being SA and SB.

and Stream II. with r of the stars, their motions relative

Since C is the centre of
mass of the whole. CB :
CA -- 2 : 3, and CM, CB
are the motions of the
two streams freed from
the solar motion. In the
second diagram we have
regrouped the stars into
three roughly equal streams,
the motions of which rela-
tive to the Sun are SA',
SB', SO] their absolute
motions are CA' , CB f

Three Drifts

Two Drifts

FIG. 16. Comparison of Two-drift
and Three-drift Theories.

and zero, and here CA ' is
approximately equal to
CB'. Clearly also A'B'
must be greater than AB, for in removing the slower
moving members of A and B to form a new group at C,
we increase the relative mean velocity of the remainder.

The third drift has little or no motion relatively to
the mean of the stars ; it may be considered to be
practically at rest in space. This characteristic is a
well-known feature of the stars of the Orion type,
which are usually removed from the data, in making
investigations of the two star-streams, because they are
found not to participate in the drift motions.* It is
natural to associate Drift with the Orion stars as
sharing the same peculiarity of motion ; but it is not
quite clear how close is the relation between them.

* The Orion star^ \\viv removed from the writer's investigation of Boss's
Preliminary General Catalogue, but not for this reason, which was in fact
unknown at that time. Their tendency to form vague moving clusters was
the main objection; further, as tin.- motions of those not included in the
clusters is extremely small and, therefore, the direction of the motion is
inaccurately determined, it was thought that the value of the results would
In- improved by their removal (M<,nthli/ A"o//V/.s, vol. 71, p. 40).

THE TWO STAR STREAMS

123

The elements of the three drifts determined by Halm
from the Boss proper motions are :

Apex.

Speed.
Drift I. .

Drift II. .
Drift O .

R.A.

90
27CP

90'

Dec.
J
-49
-36

fcF-0-9

From the analogy between the Drift and the Orion
type stars, Halm was led to assume that their internal
motions are smaller than those of the other drifts ; thus

H

Drift I.

FIG. 17. Distribution of Drifts along the Equator (Halm).

the speed given above is measured in terms of a different
unit //.

The additional number of constants to be determined
in the three-drift analysis makes the results from
individual regions very uncertain. Great variations in
the relative proportions of the stars in the drifts were
found ; these are partly accidental owing to the un-
certainty of the analysis, but are doubtless also partly real.
The diagram (Fig. 17) shows the distribution of the
stars along the equator in different right ascensions, the
radius drawn from S showing the number of star>
belonging to the drift. The apparent relation between

I2 4

STELLAR MOVEMENTS

CHAP.

Drift and Gould's belt of bright stars is probably
due to the fact that many of the brightest stars belong
to the Orion type.

Following the attitude we have adopted with regard
to the other two hypotheses, we may be content to
regard the three-drift theory as only an analytical
summary of the distribution of stellar motions, without any
hypothesis as to the physical existence of three separate
systems. There is, however, an important property of
Drift 0, which seems to make it something more than
a mathematical abstraction. We have seen that all the
stars of the Orion type appear to belong to it, and not
to the two other drifts. It is true that in addition it
contains stars of the other types, which are not in any
way distinctive ; but that it should contain the whole
of one spectral class seems to show that it corresponds
to some real physical system, and places it on a some-
what different footing from the two older drifts, for
which we have as yet failed to find any definite
characteristic apart from motion.

SUMMARY OF DETERMINATIONS OF THE CONSTANTS OF THE Two STREAMS.

True Vertex.

Ref.

No.

Catalogue or
Data used.

Investigator.

Vertex.
R.A. Dec.

Hypothesis.

1
2
3
4
5
6
7

8

Auwers-Bradley . . .

,, ...

*
Groom bridge ....
....
Boss
*5

Zodiacal . ...

Kapteyn . . .
Rudolph . .
Hough and Halm
Eddington . . .
Schwarzschild .
Eddington . . .
Charlier ....

Eddington . . .

C

91 +13
96+7
90+8
95+3
93+6
94 +12
103 +19

109 + (>

Two-stream.
Ellipsoidal.
Two-drift.
Two-drift.
Ellipsoidal.
Two-drift.
Generalised
Ellipsoidal.
Two-drift.

9
10
11
12

Large Proper Motions .

Radial Velocitu * . .
Boss . .

Dyson ....
Beljawsky . . .
Hough and Halm

88 +21

86 +24
88 +27
not given

Two-drift.

Ellipsoid il.
Two-drift.
Two-stream.

13

Faint stars (7 m '0 - 13 m ).

Comstock . . .

87 +28

Ellipsoidal.

VI

THE TWO STAR STREAMS

125

With regard to these it may be remarked that the
method used in No. 7 gives greatest weight to the nearest
stars, so that Nos. 7, 9, 10 refer mainly to the stars closest
to our system. No. 10 is somewhat tentative, as the
analysis used is not rigorously applicable to stars selected
on account of large proper motions. No. 8 is probably
affected by a systematic error in the data used. No. 1 1
would probably be revised, if account were taken of the
systematic error now believed to exist in the determina-
tions of radial velocity of the Orion type stars.

Apices of the Tn:o Drifts.

Ref.
No.

Catalogue.

Investigator.

Drift I.

Drift II.
R.A. Dec.

1 Auwers-Bradley . . . Kapteyn . . .
3 ,, ... Hough and Halm

4 Groombridge

Eddington . . .

85 -11
87 -13

90-19

260 -48
276 -41
292 -58

6 Boss Eddington . . . ! 91 -15 288 -64

12 ,, Boss 96-8 290-54

9 Large Proper Motions . Dyson . . . . j 93 - 7 246 - 64

For the velocities of the two drifts relative to the sun
the results are :

No. 3 ratio 3 : 2

,,4 1-7 and 0'5

,,6 1-52 0-86

,,9 ratioS : 2

For the ratio of the minor axis to the major axis of the
Schwarzschild ellipsoid, the determinations are :

So. 2 ....... 0-56

,,5 0-63

,,7 0-51

,,10 0-47

13 . 0-62

126

STELLAR MOVEMENTS

CH. VI

No. 1 Kapteyn . . .

,, 2 Rudolph . . .

., 3 Hough and Halm

,, 4 Eddington . . .

,, 5 Schwarzschild .

,, 6 Eddington . . .

,, 7 Charlier ....

., 8 Eddington . . .

,, 9 Dyson ....

,, 10 Beljawsky . . .

,, 11 Hough and Halm

,-, 12 L. Boss

13 Comstock . . .

AUTHORITIES.

British Association Report, 1905, p. 257;

Monthly Notices, 72, p. 743.
A sir. Nach., No. 4369.
Monthly Notices, 70, p. 568.
Monthly Notices, 67, p. 34.
Gb'ttingen Nachrichten, 1907, p. 614.
Monthly Notices, 71, p. 4.

Lund Observatory k Medd.elanden,' Serie2, No. 9.
Monthly Notices, 68, p. 588.
Proc. Roy. Soc. Edinburgh, 28, p. 231, and 29,

p. 376.

A sir. Nach., No. 4291.
Monthly Notices, 70, p. 85.
Unpublished, but see B. Boss, Astron. Journ.,

No. 629.
Astron. Journ., No. 655.

REFERENCES. CHAPTER VI.

1. Kapteyn, Address before St. Louis Exposition Congress, 1904.

2. Eddington, Monthly Notices, Vol. 67, p. 34 (Region F).

3. Eddington, Monthly Notices, Vol. 71, p. 4.

4. Boss, Astron. Journ., No. 614.

5. Schwarzschild, Gb'ttingen Nachrichten, 1907, p. 614.

6. Dyson, Nature, Vol. 82, p. 13.

7. Eddington, Monthly Notices, Vol. 69, p. 571.

8. Dyson, Proc. Roy. Soc. Edinburgh, Vol. 28, p. 231.

9. Campbell, Lick Bulletin, No. 211.

10. Eddington, Monthly Notices, Vol. 67, p. 58.

11. Eddington, ibid., p. 59.

12. Eddington, Monthly Notices, Vol. 68, p. 104.

13. Hough and Halm, Monthly Notices, Vol. 70, p. 85.
14 Hough and Halm, ibid., p. 568.

15. Kobold, Astr. Nach., Nos. 3435, 3591.

16. Eddington, Monthly Noticts, Vol. 71, p. 40.

17. Halm, Monthly Notices, Vol. 71, p. 610.

CHAPTER VII

THE TWO STAR- STREAMS MATHEMATICAL THEORY

Ix the preceding chapter we have described the principal
results of researches on the two star- streams. We shall
now consider the analytical methods used in the

investigations.

TWO-DRIFT HYPOTHESIS

Consider a region of the sky, sufficiently small to
be treated as plane, and consider the motions of
the stars projected on it. Suppose that there is a
drift of stars, i.e. a system, in which the motus peculiares
are haphazard, but which is moving as a whole relatively
to the Sun. We take, as the mathematical equivalent
of haphazard, a distribution of velocities according to
Maxwell's Law, so that the number of stars with
individual linear motions between (u,v) and (u + du,
v + dv) is

In justification of this it may be pointed out that Max-
well's is the only law T for which the frequency is the same
for all directions, and at the same time there is no corre-
lation between the x and y components of velocity.
There are other laws, which make the motions random in
direction ; but for them the expectation of the value of a
component velocity u will differ according to the value of

127

128 STELLAR MOVEMENTS CHAP.

the other component v ; for instance, with the law
a large component v is likely to be accompanied by
a large component u. Now we are not at the moment
concerned with what law stellar motions are likely to
follow ; that is a dynamical problem. We are rather
choosing a standard of comparison with which to compare
the actual distribution of motions, and that standard
ought to be the simplest possible. It is by no means
unlikely that large values of v may be correlated with
large values of u ; but, if that should be the case, we
ought to discover it as an explicit deviation from the
simpler assumption of no correlation rather than conceal
it in our initial formulae ; for it is an interesting result
that has to be accounted for. In seeking for the unknown
law of stellar velocities, we are at liberty to adopt any
standard of comparison that we please, but there is clearly
a special propriety in taking Maxwell's Law, as it is the
nearest possible approach to an absolutely chaotic state of
motion.

In the frequency-law

N is the total number of stars considered and h is
a constant depending on the average motus peculiaris.
It is related to the mean speed (in three dimensions) Q by
the equation

-W5

Let

F the velocity of the drift, taken to be along the axis of x ;

r = resultant velocity of a star ;

& = position angle, or inclination to Ox, of the resultant velocity

(The velocities are all to be taken in linear, not angular
measure.)
Then

u t + tf = v &+ V*-2
dudv = r dr dti.

VII

THE TWO STAR STREAMS

129

Thus the number of resultant motions between position
angles 6 and 6 4- dO is

- 2 Vr cos Wr .

Setting
the number is

Writing

the following table gives the values of log/(T)

TABLE

15.

The Function f

.

T.

log/(r).

T.

log/(r).

r.

log/(r).

T.

log/(r).

-1-2

1-0411

-0-3

1-5363

0-5

0-1876

1-3

1-1520

-1-1

1-0874

-0-2

1-6046

0-6

0-2886

1-4

1-3003

-1-0

1-1355

-o-i

1-6763

0-7

0-3947

1-5

1-4555

-0-9

1-1856 0-0

1-7514

0-8

0-5061

1-6

1-6177

-0-8

1-2378 0-1 1-8303

0-9

0-6232

1-7

1-7871

-0-7

1-2923 0-2 1-9131

1-0

0-7461

1-8

1-9637

-0-6

1-3493 0-3

o-oooi

1-1

0-8751

1-9

2-1478

-0-5

1-4088

0-4

0-0916

1-2

1-0103

2-0

2-3393

-0-4

1-4711

For a single drift the number of stars moving in any
direction 6 is proportional to f (h V cos 6} ; and the
equation to the theoretical single-drift curves discussed
on p. 88 is

r oc /(/iFcostf).

The usual method of analysis is to compound two such
curves pointing in different directions, adjusting the

1 30 STELLAR MOVEMENTS CHAP.

various parameters by trial and error until a satisfactory
approximation to the observations is obtained.

A mathematical method of determining the double-drift
formula

r = a l f(hV l 0080-0!) + a 2 /(/iF 2 cos 6- 2 ),

which best represents the observations, without recourse
to trial and error, has been given. 1 It worked quite
satisfactorily for the Groombridge regions, where the stars
were very numerous ; it is not, however, to be recommended.
A mechanical method, which automatically gives some
sort of answer, whether the distribution really corresponds
to two drifts or not, is not so discriminating as the simpler
synthetic process.

ELLIPSOIDAL HYPOTHESIS.

It has been seen that the principal fact of the
phenomenon of star-streaming is the greater mobility of
the stars along a certain line than in the perpendicular
directions. K. Schwarzschild represents this mobility by
assuming that the individual motions are distributed
accordin to the modified Maxwellian law

where, k being less than h, the u components of velocity
are on the average greater than the v and w components.
In two dimensions, let the number of stars with individual
motions between (u, v) and (u + du, v + dv) be

And let the components of the parallactic motion of
the whole system be (U, V). The parallactic motion is
not in general along the axis of greatest mobility Ox.

As before, let r, 6 be the magnitude and direction of
the resultant velocity of a star. Then

+ TiV = k\r cos 6 -17)* + W(r sin 6 - F) 2 ,
dn do = r dr cW.

vii THE TWO STAR STREAMS 131

The number of stars moving in directions between
and 6 + dd

_ IThk^Q r r r e -r2(fc2 C os 2fl+^sin2)+2r(fc2D-cos 0+ft2p s in 0) - fc2[j2_ 7,272
7T Jo

Setting

t _

J*
ac=rVjp-

The number becomes

Nhk -WV 2 -^ [">

JO

-v** e f r e-

p J -

The integral leads to the same function / as before ; and
the number of stars moving in any direction is proportional
to

A little consideration shows that the polar curve
r= - / (f ) will closely resemble a two-drift curve.

f is a maximum near the direction of the parallactic motion
( U, V) and a minimum in the opposite direction ; and the
same is true for /(). This factor alone would give a
curve not very different from a single drift curve. But

the factor corresponds to an ellipse with its major axis

along Ox. It distorts the approximately single-drift
curve, pinching it along Oy and extending it along Ox,
with the result that a bi-lobed curve is usually obtained.

The method of determining the constants of an ellip-
soidal distribution to suit the observations may be briefly
noticed.* If we consider the stars moving in a direction
6 and in the opposite direction 180 4- 6, p is the same for
both directions, and simply changes sign. Thus the

* I have made slight alterations in the procedure given by Schwarzschild
in order to conserve the close correspondence with the two-drift analysis.

K 2

132

STELLAR MOVEMENTS

CHAP.

ratio of the numbers of stars moving in these directions
gives -J-f Y From the table of log/, we construct the

following table (the logarithm is given as being more
convenient for interpolation) :

TABLE 16.
Auxiliary Function for the Ellipsoidal Theory.

6

/tf>.

6

*k

0f V(-).

o-o

o-ooo

0-5

0-779

o-i

0-154

0-6

0-939

0'2

0-309

07

1-102

0-3 0-464

0-8

1-268

0-4 0-620

1

This enables f to be found from the observations, and,
when it is known, p is given by

number of stars = -/()

If now we take radii r^ = and r 2 =

in the direc-

tion 0, r will trace out the ellipse

and r. 2 the straight line

By drawing the best ellipse and straight line through
the respective loci, & 2 , A 2 , U and F are readily found ; also
the direction of greatest mobility, which is the major axis
of the above ellipse, is determined.

A very elegant direct method of arriving at the values
of these constants has also been given by Schwarzschild - ;
it appears, however, to be open to the same objection as
the automatic method of determining the constants of the
two star-streams. If these methods are used at all, it
is very necessary to examine afterwards how closely the

THE TWO STAR STREAMS

133

solutions represent the original observations. I have
sometimes found them to be very misleading.

The ellipsoidal hypothesis has been applied with great
success by Schwarzschild to the analysis of the proper
motions of the Groombridge catalogue. As an illustration
we may take the region R.A. 14 h to 18 h , Dec. +38 to
+ 70 already considered on the two-drift theory (p. 90).
The comparison of the two hypotheses with the observa-
tions is given in Table 17. In only two lines do the two
representations differ from one another by more than a
uni t.

TABLE 17.

Comparison of Ellipsoidal and Two-drift Hypotheses ivith Observations.

T\'

Number of Stars.

"r^i %.y*rt

Number of Stars.

Direc-
tion.

Ob-
served.

Ellip-
soidal.

Two

Drift.

.Direc-
tion.

.Ob-
served.

Ellip-
soidal.

Two

Drift.

5

4

5

6

205

21

22 22

15

5

6

7

215

27

25

26

25

6

7

8

225

29

26

27

35

9

9

10

235

26

27

26

45

10

11

11

245

19

23

22

55

14 12

12

255

17

18

18

65

14 13

12

265

12

14

14

75

14 14

13

275

11

10

10

85

13 13

13

285

11

8

95

12 13

12

295

8

6 6

105

10 12

13

305

7

5 5

115

11 11

12

315

6

5 4

125

10 10

11

325

6

4

5

135

10 10

9

335

5

4 5

145

7 10

9

345

5

5

6

155

9 11

9

355 4 5 6

165

9

12

11

175

14

14

12

185

14 16

15

195

16 19

19

Whilst the two hypotheses can yield closely similar
distributions of motion as regards direction, it is conceiv-
able that if the magnitudes of the motions were taken
into account the resemblance might fail. But it is not

134

STELLAR MOVEMENTS

CHAP.

difficult to see that the two laws express distributions of
linear velocities very similar in most respects, though by
the aid of different mathematical functions, provided that
the numbers of stars in the two drifts are practically
equal. Schwarzschild's method is somewhat analogous to
replacing two equal intersecting spheres by a spheroid. If
we have two equal drifts, with velocities + V and V
referred to the centre of mass of the whole, the frequency
of a velocity (u, v) is proportional to

e -

or to

The ellipsoidal law may be written

The difference in the two laws is thus determined by the
difference between cosh au and e 13 " 2 , functions which have
a considerable general resemblance.

As both laws give the same distribution of the v com-
ponents, we may confine attention to the u components.
As a typical example for comparing the laws (corresponding
approximately with what is actually observed), we will

k I

take h V= 0'8, ^ = 0*58, ^ = 20 km. per sec.

TABLE 18.
Comparison of Two-drift and Ellipsoidal Hypotheses.

Component

Frequency.

Difference

u.
Km. per sec.

Two-drift
Hypothesis.

Ellipsoidal
Hypothesis.

Ellipsoidal -
Two-drift.

1-055

1*160

+ 0-105

10

1*099

1-066

-0-033

20

1-000

0-828

-0-172

30

0-618

0-544

-0-074

40

0-237

0-302

+0-065

50

0-056

0-141

+ 0-085

60

0-008

0-056

+ 0-048

70

o-ooi

0-019

+ 0-018

VII

THE TWO STAR STREAMS

The two curves are shown in Fig. 18.

Whilst there is a marked general resemblance, the
differences are not altogether negligible. In particular
the ellipsoidal law gives a considerably greater number
of large velocities ; it seems probable that in this respect
it is better fitted to the observations. The two-drift
law gives a defect of both very small and very large
motions, as compared with the simple error law ; this
is sometimes expressed by saying that it has a negative
excess.

It may further be noticed that, although in the
example given the frequency of the two-drift distribution

\

-80

-60

-40

40

60

80

-20 20

Km. per sec.

FIG. 18. Comparison of Two-drift and Ellipsoidal Hypotheses.
Two-driftfull curve ; Ellipsoidal dotted curve.

makes a slight dip at the origin u = 0, for rather
smaller values of hV this dip disappears, and the
distribution actually agrees with the ellipsoidal distribu-
tion in having a maximum at the origin. 3

When the restriction that the two drifts have equal
numbers of stars is removed, the* ellipsoidal hypothesis
cannot approximate to the two-drift hypothesis so closely.
There is no longer a fore-and-aft symmetry, so that the
ellipse is an unsuitable figure to represent the frequency.
The two-drift theory, having one additional disposable
constant, is now able to give a considerably better
representation of the observations. We have seen that
the Groombridge proper motions can be represented by

136 STELLAR MOVEMENTS CHAP.

two drifts with approximately equal numbers of stars ;
these can be replaced by an ellipsoidal distribution with
practically the same precision. The Boss proper motions,
on the other hand, require a mixture of the drifts in
the proportion of about 3:2; the ellipsoidal hypothesis
cannot be adapted to this skewness and accordingly fails
to represent these observations. For this reason it has
not been possible to analyse these more recent proper
motions on Schwarzschild's theory.* On the other hand
the main teaching of that theory remains, viz., that the
dissection into two drifts may be only a mathematical
procedure, and that it is possible to regard the distribu-
tion of velocities as one whole.

COMBINATION OF RESULTS FROM DIFFERENT REGIONS OF

THE SKY.

In the case of the two-drift theory the procedure is
very simple. Let X^ Y^ Z l be the components of the
velocity in space of one of the drifts, measured in the
usual unit l/h ; v 1 and 0,, its velocity and position angle,t
determined for a region whose centre is at (a, 8). We
have for each region equations of condition

t

v t sin &i X^ sin a + Y l cos a,

v l cos 6 l = - X l cos a sin 6 - Ysin a sin d + Z^ cos fi,

from which X lt Y lt Z^ can be found.

On the ellipsoidal hypothesis the projected solar motion
is found for each region, and the results can be combined
in the same way. But the combination of the ellipsoidal
constants is a more complex problem.

It will be useful to take the general ellipsoid with three
unequal axes, although in ordinary applications a spheroid

* C. V. L. Charlier has developed a generalisation of the ellipsoidal theory
which permits the skewness to be taken into account, but his method
assumes some knowledge of the distribution of the stars in space.

f The position angle is here measured from the meridian.

vii THE TWO STAR STREAMS 137

only is considered. Keferred to any rectangular axes, let
the velocity -ellipsoid be

>iii- + l>\r + civ 2 + 2fvw + 2givu + 2huv = 1,

so that the number of stars with individual velocities
between (it, v, iv) and (u + du, v + dv, tv + div) is propor-
tional to

Take the w direction to be the line of sight. To obtain
the distribution of the projected velocities (u, v) we must
integrate the above expression with respect to w from
oo to + oo . The result is

exp - au* + 6t- 2 4- 2huv - . du dv

The integral in this expression is a constant and equal

,0

Thus the projected velocities correspond to a velocity-
ellipse

Xow this ellipse is the right-section of the cylinder
parallel to the i^-axis, wliich passes through the inter-
section of the ellipsoid

au 2 + 6r 2 + cw- + 2fcw + 2gwu + 2huv = 1,

and the plane

The latter is the diametral plane conjugate to the
zr-axis, and consequently the cylinder is the enveloping
cylinder.

Thus the velocity-ellipse for any region is simply the
outline of the velocity-ellipsoid viewed from an infinite
distance in the corresponding direction. The outline
must, of course, not be confused with the cross-section
which is a different ellipse.

138 STELLAR MOVEMENTS CHAP.

Now let the velocity- ellipsoid, transformed to its princi-
pal axes, be

and the line of sight be in the direction (I, m, n). The
lengths of the axes of the enveloping cylinder (i.e., of the
velocity-ellipse) are given by

72 /w2 2

+ m + n = o,
and the direction ratios of these axes are then

It may be noted that there will be four points of the
sky for which the velocity-ellipse will be a circle, and the
projected motions will be haphazard. When the ellipsoid
is a spheroid these points coalesce with the two extremities
of the axis, in other words with the vertices.

The general case of the ellipsoid with three unequal
axes is of considerable interest, because it enables us to
allow for the possibility that the motions may have a
special relation to the plane of the Milky Way as well
as to the axis of star-streaming. The fact that stellar
motions have some tendency to be parallel to the Milky
Way was pointed out by Kobold, and in recent years by
the investigators of radial velocities. Since the stellar
system is strongly flattened towards this plane, the result
seemed a very natural relation between motion and
distribution. But the star-stream investigations have
shown that the main tendency is not towards a general
parallelism to the Milky Way but a parallelism to a
certain direction in it. Whether there is any residual
relation to the Milky Way not covered by this, is a matter
of some interest. We could test it by making an analysis
on the basis of a velocity-ellipsoid with three unequal
axes, and noticing whether the two smaller axes turn out

vii THE TWO STAR STREAMS 139

to be equal to one another. The difficulty is that, as
already stated, the ellipsoidal hypothesis does not give a
satisfactory representation of Boss's proper motions, which
would naturally be used for such a test. It seems, how-
ever, fairly certain that the deviation from a spheroid
must be very slight. The evidence from the radial
motions (Table 9) points in the same direction.

Taking the velocity-ellipsoid to be a prolate spheroid

7,V'* 4- hftv* + to 2 ) = 1

and the velocity-ellipse for a region, with centre at an
anular distance % from the vertex, to be

then, since the velocity-ellipse is the apparent outline of
the ellipsoid, we have

_ ,

F = ~kf V

and

and hence

Thus the minor axis y of the velocity-ellipse is the same

throughout the sky ; and the last equation expresses the
variation of the major axis.

Also the major axis is directed along the great circle to
the vertex.

MEAN PROPER MOTIONS.

These have been used in the previous chapter to
determine the mean distances of the two drifts. It
would have been possible to start with the mean proper
motions in the different directions, instead of the
simple frequency, as data, for the purpose of exhibiting
and analysing the star-stream phenomena. But, besides
being much less sensitive, there is the objection noticed

1 40 STELLAR MOVEMENTS CHAP.

in Airy's method of finding the solar motion, that
an average motion is likely to depend mainly on a
few specially large values, and is very much subject to
accidental fluctuations. To use the frequency of proper
motions leads to much smoother results ; and further, as
we avoid giving excessive weight to the nearest stars, the
results should be more representative of the stars as a
whole. It is therefore better to reserve the mean proper
motions for obtaining new information not deducible from
the frequencies.

We found that on the ellipsoidal hypothesis the number
of stars moving in directions between 6 and 6 + d9 was
proportional to

P -f
Hence for the mean value of x

The numerator of this expression is equal to

The integrated part vanishes at both limits. We thus obtain

But = r
Hence if

the mean linear motion in any direction is

VII

THE TWO STAR STREAMS

141

For a simple drift *Jp reduces to h, and % to h V cos 0,
so that the mean linear motion is

The values of g are given in Table 19.

TABLE 19.
The Function y (r).

T.

<j(r).

T.

<7(r)-

1 '

*.

-1-0

0-565

-o-i

0-845

0-8

1-315

-0-9

0-589

o-o

0-886

0-9

1-381

-0-8

0-614

o-i

0-930

1-0

1-449

-07

0-641

0-2

0-977

1-1

1-520

-0-6

0-670

0-3

1-027

1-2

1-594

-0-5

0-701

0-4

1-079

1-3

1-669

-0-4

0-734

0-5

1-134

1-4

1-747

-0-3

0-768

0-6

1-191

1-5

1-827

-0-2

0-805

07

1-252

1-6

;

1-908

For determining the distances of the two drifts, equations
of condition are formed as follows :
Let

r be the mean proper motion in the direction 6.
rfj, d. 2 the unknown mean distances of the stars of the two drifts

(i.e. distances corresponding to the mean parallaxes)
n lt 7i. 2 the numbers of stars of the two drifts moving in a direction 6.
These have been determined by the previous analysis of the
directions of motion.
F,, 0, ; Vo, Oo the velocities and directions of the drift motions

Then

l cos 6-6^} -- + n^(A F 2 cos - 2 ) --

Forming these equations of condition for successive values

of 0. we can determine -j-r- and ^ =- by a least-squares

hd l hd 2 J

solution.

On the ellipsoidal hypothesis there is only the one
unknown d, the mean distance of the stars, to deal with.
It can be determined by the equations of condition

mean proper motion in direction 6 = - . a().

d J

142 STELLAR MOVEMENTS CHAP.

Or the mean proper motions may be used to effect an
independent determination of the ellipsoidal constants
exactly as the frequency was used. The latter procedure
is much simplified by the aid of the following theorem :

If in the direction 6 a radius is taken, which is the
geometric mean between the mean proper motion in
the direction 6 and the mean proper motion in the
opposite direction, the radius will trace out the velocity-
ellipse (within an extremely small margin of error).

The mean proper motions in the directions 6 and 6+ 180
are respectively

* riband -Lrf-0.

N/JP *JP

Now f can never be as great as the ratio of the solar
motion to the minor axis of the velocity-ellipsoid.
Actually 0'5 is about the upper limit, but to allow an
ample margin we shall take also 1*0. From Table 19,

for | =0-0 */KGf(-D = 0-8862
0-5 0-8914

1-0 0-9049

Thus ^/g(^) g( f ) may be taken to be constant with an
error not greater than one in fifty in the most extreme
case. The geometric mean of the mean proper motions

is then proportional to r , which is the radius of the

velocity-ellipse in the corresponding direction.

The theorem provides a short method of finding the
velocity-ellipse in any part of the sky, provided a fairly
large number of observed proper motions are available.
It is subject, however, to the drawback that there are
generally some directions in which very few stars are
moving, so that we have to take the geometric mean of
two quantities, one of which is badly determined and the
other unnecessarily well determined. For the numerous
motions of the Groombridge Catalogue the method proved

VII

THE TWO STAR STREAMS

satisfactory, and the results agreed closely with those
found from the simple frequencies of the proper motions. 4

TABLE 20.
Groombridge Proper Motions.

Ratio of Axes of Velocity-Ellipse.

Region.

No. of Stars.

Mean Proper
Motions.

Frequency of
Proper Motions.

A

585

0-59

0-59

B

862

0-56

0-58

C

516

076

0-70

D

443

0-82

0-81

E

385

0-65

0-72

F

425

0-53

0-61

G

1103

0-66

072

RADIAL MOTIONS TWO-DRIFT HYPOTHESIS.

The development of the formulae necessary for the
study of radial motions will now be considered. The radial
differ from the transverse motions in two respects, (l) The
transverse motions allow us to compare the motions in
two perpendicular directions in the same region of the
sky ; but to learn anything as to the form of the velocity
distribution from the radial motions it is necessary to com-
pare the results from different regions. This is evidently
a disadvantage, for it introduces the complication of the
differences between galactic and non-galactic stars, local
drifts, and so on. (2) The results come out in linear
measure, independent of the distances of the stars.

It will be assumed throughout that the radial velocities
have been corrected for the solar motion, and are accord-
ingly referred to the centre of mass of the system.

The effect of the preferential motions in the directions
of the two vertices will be that the radial velocities will

i 4 4 STELLAR MOVEMENTS CHAP.

be greater on the average near the vertices than in other
parts of the sky. This was illustrated in Table 9 (p. 107)."

Taking first the two-drift theory, let V l and F 2 be the
velocities of the two drifts, referred to the centroid of the
whole system ; a and 1 a the proportion of stars in each
drift.

Then

aF, = (l-a)F 2 .

The mean radial velocity regardless of sign near the
vertices is for Drift I.

and the mean radial velocity at right angles to the
vertices is

l

hjir'

Thus for the two drifts the average radial velocity at the
vertex is to the radial velocity in a region 90 from the
vertices in the ratio

If, using the results of the analysis of Boss's Catalogue, we
set

a = 0-6 l-a = (H

7^=075 hV 2 = 1-12,

the ratio becomes 1727

In Table 9 the mean observed ratio was

15'9 km. per sec. _ j.
9*5 km. per sec.

and, allowing for the large size of the areas considered, the
agreement is remarkably exact. But the confirmation is
not quite so satisfactory as it appears at first sight, because
the stars of Table 9 are of Type A, a type showing the
star-streaming very strongly ; and there is no doubt that

vii THE TWO STAR STREAMS 145

if Type A alone had been used in the transverse motions
higher values of hV l and hV. 2 would have been obtained.
According to H. A. Weersrna 5 the drift-velocities for Type
A are

the probable error being, however, nearly 10 per cent.
These numbers lead to the ratio 2*02.

Considering the uncertainty both of the observed ratio
and of the drift-constants for Type A, the discordance
between 1*68 and 2*02 is not unduly great.

RADIAL MOTIONS ELLIPSOIDAL HYPOTHESIS.

Generally speaking, Schwarzschild's ellipsoidal hypo-
thesis is the most convenient for the mathematical
discussion of the radial motions. The first problem is
to find the distribution of the radial velocities at a
particular point of the sky in terms of the axes of the
velocity-ellipsoid. It must be noticed that the distribu-
tion of the component-velocities in any direction is by
no means the same as the distribution of whole velocities
in that direction.

Let the velocity-ellipsoid referred to its principal
axes be

U 2 V 2 1V* _ 1
+ r: 7 "T o J-

- b- c*

and let the line of sight be in the direction (, m, n).

Referring the ellipsoid to three conjugate diameters
two of which, a' ', V, are in the plane perpendicular to
(/, m, n) the equation can be written

u' 2 v" 2 , w' 2 _ ,

^ + p + c , 2 -

and the frequency of the oblique velocity components
w' is proportional to

If now V be the rectangular velocity component in the

L

146 STELLAR MOVEMENTS CHAP.

line of sight, p the perpendicular on the tangent plane
normal to the line of sight,

w' ^~c'

Thus the frequency of a component V in the direction
/, m, n is proportional to

<
that is

The fact that the divisor is the perpendicular on
the tangent plane is analogous to the two-dimensional
result that the velocity-ellipse is the right section of the
tangent-cylinder.

To determine the velocity-ellipsoid from a series of
measures of radial velocities, suppose first that the
observations are approximately uniformly distributed over
the sky. By the foregoing paragraph the mean value
of F 2 in any part of the sky is proportional to
a 2 / 2 + 6 2 m 2 + cV, or, referred to more general axes, to a
homogeneous expression of the second degree in Z, m, n,
say E.

Form now from the observed data the coefficients

A = SF 2 Z 2 F

B = ZFW G

C =

Then

A\* + B + O> 2

This last expression is the moment of inertia of the

surface r' L = E* about the plane whose normal is (\, /*, v).

The surface is not the velocity-ellipsoid, indeed it is

not an ellipsoid at all ; it is the inverse of the reciprocal

* The mass beini? supposed to be distributed proportionately to the solid
angle, or, more strictly, proportionately to the numbe|: of observations of
radial velocity.

vii THE TWO STAR STREAMS 147

ellipsoid. But it is evident that it will have the same

principal planes as the velocity-ellipsoid. Thus the

directions of the axes of the velocity-ellipsoid are those
of the momental ellipsoid of this surface, that is of the
quadric

2Gv\

These are given by the direction -ratios
i i i

GH-F(A-k) ' HF-G(B-k} ' FG-H(C-k),

where k has in succession the values of the three roots of
the discriminating cubic

A-k H G =0.

H B-k F
G F C-k

Since the surfaces r 2 = E and r 2 = *JE have the same
principal planes we may use | F | instead of V 2 in form-
ing the coefficients, so that

A = 2 | V | I 2 , F = 2 | F mn, etc.

This is probably a preferable procedure, since squaring
the velocities exaggerates the effect of a few exceptional
velocities ; just as in calculating the mean error of a series
of observations it is preferable to use the simple mean
residual irrespective of sign rather than the mean-square
residual.*

When the observed radial velocities are not uniformly
distributed over the sky the problem is more complex, but
there is no great difficulty in working out the necessary
formulae.

EFFECT OF OBSERVATIONAL ERRORS.

The accidental errors in the determination of the proper
motions must tend to equalise the number of the stars
moving in the different directions, and to smooth out the

* This is contrary to the advice of most text-books ; but it can be shown
to be true.

L 2

I 4 8 STELLAR MOVEMENTS CHAP.

peculiarities of distribution caused by star-streaming. In
consequence, the deduced velocities of the two drifts are
likely to be too small. An approximate calculation of
the amount of this effect may be made by taking the
simple case when the stars are all at the same distance
from the Sun.

In this case the accidental errors of the proper motions
will simply reappear (multiplied by a constant) as the
accidental errors of the linear motions. If the true
frequency of a component of linear motion u be

and the frequency of an error x in it be

the apparent frequency of linear motion u will be

du r J^e-^-^.^-e-^dx
J -oo VTT \ /7r

du TOO hk _(h2+&) X 2+2hVux-h*u2 j

= 7= / = e ax

V7T J -

00 V7T

Jilf fOO / H 2 u \2

* ^L / *-<*+*!( ~ J3?p) dx

VTT-' -oo

The true frequency distribution with the constant h is
thus replaced by an apparent distribution with a constant
Aj where

_!_ = !+!

V h* fc2*

To apply this formula we may use the determinations
of j-j which have been made for several regions.
Thus taking Region B of Groombridge's Catalogue

= 2" '4 per century.
ha

vii THE TWO STAR STREAMS 149

The probable accidental error of a Groombriclge proper
motion is about 0"'7 per century.

Therefore

0-477 0"7 ,, l
~k~ ~~ ^T h

1 0-61

k

^ I + (0-61) 2

h? /i 2

J_ 1>17

^ h

Thus for Region B the deduced velocities of the drifts
need to be increased in the ratio .

For the Boss proper motions the correction is not
so important. Taking the large region already discussed
(p. 116)

= 7"'2 per century.
ML

The probable error of a Boss proper motion = 0"'55 per
century.

0-477 0-55 l

~k~ W h

1 0-160

/; h

_!_ 1-013

^ h

The correction is only about one per cent.

The hypothesis that the stars are at the same distance
is very far from true, and in consequence the corrections
here determined are only rough; but the calculation is
sufficient to show that when the motions are small and
not very well determined the effect of the accidental errors
may be quite appreciable.

Of the possible systematic errors, the most important
are those due to an error in the adopted constant of
precession and an error in the adopted motion of the
equinox. The former would lead to an apparent rotation
of the stellar system about the pole of the ecliptic ; the

1 5 o STELLAR MOVEMENTS CHAP.

latter to a rotation about the pole of the equator. There
is no way of determining the constant of precession
except by a discussion of stellar motions ; the motion of
the equinox can, however, be determined from a discussion
of observations of the Sun, and it is a matter for consider-
ation how much weight ought to be attached to the solar
and stellar determinations, respectively. The recognised
determinations of these two constants have been based on
the principle of haphazard velocities ; on the two-stream
theory the solution would become extremely difficult, and
to some extent indeterminate, if the velocities and propor-
tions of mixture of the two streams are not exactly the
same all over the sky. The same difficulty occurs in
defining the constant of precession as in defining the solar
motion ; though in this case the difficulty is a practical
and not a philosophical one.* In practice it is, within
reasonable limits, arbitrary how much of the observed
motions shall be attributed to the rotation of the axes of
reference, and how much to the heavenly bodies themselves.
The only guide is that the residual stellar motions should
follow as simple a law as possible. But, when it is certain
that no really simple law is possible, this is not a condition
that can be expressed and used analytically. When once
the hypothesis of haphazard motions is given up, the con-
stant of precession can only be given an approximate
value, and there is no very satisfactory way of improving
itt

An investigation by Hough and Halm throws important
light on the relation between star-streaming and the pre-
cession-constant. It is shown by them that the inequality
of mixture of the two drifts in different parts of the sky

* In dynamics we know precisely what we mean by absolute rotation
though we may not be skilful enough to detect it ; absolute translation
cannot even be denned.

f A determination from the stars of the Orion type, which are supposed
to be moving at random, would be of interest, but the inequality of
distribution and the prevalence of moving clusters would make it difficult.

VII

THE TWO STAR STREAMS

accounts for certain discordances in former investigations
of the precession. But this work does not appear to lead to
any means of determining the constant de novo.

Having regard to the practical impossibility of obtaining
an accurate value of the precession-constant, there is an
important advantage in proceeding so as to avoid the
systematic errors arising therefrom. This is done when
we treat two antipodal areas of the sky together ; for the
error of precession will be in opposite directions (in space)
in the two areas, and its effect will be wholly or partially
eliminated in the mean result.

THE MAXWELLIAN LAW.

The Maxwellian or error-law plays an important part in
the analysis both of the two-drift and ellipsoidal theory.
The radial velocity determinations now available enable us
to test in a direct manner how far the stellar motions obey
this particular law.

For Type A the following table (21) compares the
actual distribution of the radial motions (corrected for
the solar motion) with an error-law. 7 In order to get
rid of most of the effect of star-streaming, stars in the
neighbourhood of the vertices, forty in number, were
not used.

TABLE 21
Radial Motions of Type A.

Number of Stars.

Limits of Velocity.

Error Law.

Observed.

km. per sec.

0-0 4-95

53-4

55

4-95 9-95

46-2

47

9-9515-95

'38-3

30

15-95 25-5

27-4

30

25-5 40

67

10

>40

0-2

152

STELLAR MOVEMENTS

CHAP.

A similar table (22) is given for Types II. and III.
(F5 M). The distribution of observed velocities has
been taken from a table given by Campbell. 8 No allow-
ance is made for the effect of star-streaming, but that
will not have so much influence proportionately as on the
smaller motions of Type A.

TABLE 22.
Radial Motitms of Types F5 M.

Number of Stars.

Limits of Velocity.

Error Law.

Observed.

km. per sec.

5

135

162

510

127

131

1015

114

124

1520

97

102

2025

78

52

2530

59

39

3035

42

33

3540

29

17

4050

30

31

5060

10

11

60-70

2

7

7080

1

4

>80

10

1

For Type A the agreement of the observed distribu-
tion of the velocities with the error-law is remarkably
close. For Types F5 M the table shows that the
correspondence is not so good. The observed distribu-
tion has what is technically called a positive excess, that
is to say, there are too many small and too many large
motions compared with the number of moderate motions.
An increase or decrease of the modulus of the error
distribution, with which it is compared, will make the
agreement better at one end but worse at the other
end of the table. A distribution of this kind would be
obtained if we mixed together error distributions having

vri THE TWO STAR STREAMS 153

different moduli ; and it may therefore be supposed
that the deviations arise from the non-homogeneity of the
material used. It is also probable that if precautions had
been taken to avoid the effects of star-streaming, the
excess of large motions would have been much less
pronounced.

"We may conclude that in a selection of stars really homo-
geneous as regards spectral type (and perhaps luminosity
also) the components of motion perpendicular to the line
of star-streaming are distributed according to the error-law,
as required by the two-drift and ellipsoidal hypotheses.
But, in a selection of stars made under ordinary practical
conditions, there is likely to be an excess of very large and
very small proper motions, and a defect of moderate
motions.

REFERENCES. CHAPTER VII.

1. Eddington, Monthly Notices, Vol. 68, p. 588.

2. Schwarzschild, Gottingen Nachricliten, 1908, p. 191.

3. Eddington, British Association Report, 1911, p. 252.

4. Eddington, British Association Report, 1909, p. 402.

5. Weersma, Agtrcfhysiedl Journal, Vol. 34, p. 325.

6. Hough and Halm, Monthly Notices, Vol. 70, p. 584.

7. Eddington, Monthly Notices, Vol. 73, p. 346.

8. Campbell, Stellar Motions, p. 198.

BIBLIOGRAPHY.

For the mathematical principles of Kapteyn's original theory, the
two-drift theory and the ellipsoidal theory, the references are :

Kapteyn, Monthly Notices, Vol. 72, p. 743.

Eddington, Monthly Notices, Vol. 67, p. 34.

Schwarzschild, Gottingen Nachrichten, 1907, p. 614.
For the direct methods, avoiding trial and error (not recommended).

Eddington, Mwtlihj Notices, Vol. 68, p. 588.

Schwarzschild, Gottingen Nachrichten, 1908, p. 191.
Other papers relating to the mathematical theory are :

Charlier, Lund Meddelanden, Series 2, No. 8 (a generalised ellipsoidal
theory, by correlation methods).

Hough and Halm, Monthly Nat !(*, Vol. 70, p. 85 (application to radial
velocities).

Oppenheim, Astr. Nach., No. 4497 (a criticism).

v. d. Pahlen, Astr. Nach., No. 4725 (a generalised theory).

CHAPTER VIII

PHENOMENA ASSOCIATED WITH SPECTRAL TYPE

IF two stars, one of Type A and the other of Type M,
are chosen at random out of the stars in space, it may
be confidently predicted (l) that the Type A star will be
the more luminous of the two, and (2) that it will have a
smaller linear velocity than the Type M star. We say
intentionally " out of the stars in space," because, for
example, the stars visible to the naked eye are a very
special selection by no means representative of the true
distribution of the stars. The odds are considerable in
favour of both predictions being correct, though a
failure may sometimes occur. Similar illustrations with
other kinds of spectra might be given. In short there
is a conspicuous correlation, on the one hand, between
spectral type and luminosity, and, on the other, between
spectral type and speed of motion. The former relation
is scarcely surprising, and some correlation would be
expected on physical grounds, although, perhaps, not so
close as that actually found ; but the connection between
type and speed is a most remarkable result.

The discovery of the latter relation has come about very
gradually. So early as 1892, W. H. Monck l pointed
out that the stars of Type II. had larger proper motions
on the average than those of Type I. Further research,
especially by J. C. Kapteyn, 2 emphasised the importance
of this discovery. It is well shown by the stars having

154

CH. vin SPECTRAL TYPE 155

excessive proper motions. In a list given by Dyson 3 of
ninety-five stars with annual proper motions of more
than 1", there are fifty-one of which the type of spectrum
is known ; of these, fifty are of Type II., and only one
(Sirius) is of Type I. Again, of those with proper motions
exceeding 0"'5, 140 belong to Type II. and four to
Type I. It was realised that this phenomenon did not
necessarily signify a connection between spectral type and
the true linear speed ; and there was a general preference
for the less startling explanation that it was due to the
feeble luminosity and consequent nearness of stars of the
second type. Certain investigations of the parallactic and
cross proper motions, as well as of the radial velocities,
appeared to confirm this view.

The next stage w r as reached in 1903, when E. B. Frost
and W. S. Adams 4 published their determinations of the
radial velocities of twenty stars of the Orion type ; it was
shown that these stars have remarkably small linear
velocities, averaging (for one component) only seven kilo-
metres per second. This result seems to have been regarded
as showing that the Orion stars w^ere exceptional ; appar-
ently it was not suspected that this was a particular case
of a general law.

With the introduction of the two-stream hypothesis, and
consequent methods of investigation, fresh light was
thrown on the subject. It was found that the " spread "
of the motions of the Type I. stars was less wide than
those of Type II., the former following much more closely
the directions of the star-streams. 5 Though other inter-
pretations were conceivable, this seemed to indicate that
the individual motions of Type I. were smaller than those
of Type II. Definite evidence was at length forthcoming
in 1910 from the results of determinations of the radial
velocities, which clearly showed that the speeds of
the second type stars were larger on the average. But
the radial velocity results led to a wider generalisation.

STELLAR MOVEMENTS

CHAP.

J. C. Kapteyn 6 and W. W. Campbell 7 pointed out indepen-
dently that the average linear velocity increases continually
as we pass through the whole series from the earliest to
the latest types, i.e., in the order B, A, F, G, K, M. The
following table contains the results of Campbell's discussion.

TABLE 23.
Mean Velocities of Stars (Campbell).

Type of Spectrum.

Radial Velocity. &<?$).

km. per sec.

B

6-52

225

A

10-95

177

F

14-37

185

G

14-97

128

K

16-8

382

M

17-1

73

Planetary nebulae

25-3

12

The velocities for F, G, and K come in the right order,
but it would be straining the figures too far to attach
much importance to this. The rise from B to A and
from A to Type II. (F, G, K), is quite well marked, and
a rise from Type II. to M is fairly indicated. The position
of the planetary nebulae at the end is distinctly curious.
If we have entire confidence in the law that the speed
increases with the stage of development, it follows that a
planetary nebula must be regarded as a final stage-
certainly not as the origin of a star. There is some
justice in a remark of K. T. A. Innes 8 : "The fact that
we have seen a star change into a nebula* ought to
outweigh every contrary speculation that stars originate
from nebulae." It is necessary to proceed cautiously in
such an application ; but we seem to have within our
grasp a new method of deciding doubtful questions as to
the order of development of the different stages in a star's
history.

* Referring to the phenomena of the later stages of a Nova.

viii SPECTRAL TYPE 157

The residual motions given in Table 23 are corrected
for the solar motion, but not for the star-stream motions.
They do not therefore represent what we consider to be
the actual individual stellar motions, as distinguished
from the systematic motions. To remove the latter
must affect the numbers appreciably. If, following
Schwarzschild's hypothesis, a is the mean speed at
right angles to the star-stream direction, and c the mean
speed towards or away from the vertex, the mean radial
speed at a point distant 6 from the vertex is

and the mean radial speed over the whole sphere is

c 2 cos 2 sin 6 dB d<j>.

This would be slightly modified by the fact that more
stars are observed near the galactic plane than in other
parts of the sky, but, since the axis of preferential motion
lies in the galactic plane, the effect of this inequality is
minimised.

Performing the integration, the mean radial speed
becomes

where

If, for example, -- =0'5G, which is probably about
c

true for the Type A stars, this speed is equal to l'30a.
So that to obtain the true motus peculiaris free from the
effects of star-streaming we should divide the result for
Type A given in Table 23 by 1'30. For the later
types the velocity ellipsoid is less prolate, and the divisor
would be smaller, about 1*15 ; Type B shows no evidence
of star-streaming and the velocity already given may

I 5 8

STELLAR MOVEMENTS

CHAP,

remain unaltered. The mean individual speeds thus
modified would then run

B, 6-5 ; A, 8-4 ; F, G, and K, 13*6 kilometres per second.

There is still a steady increase of speed with advancing
type, though the main jump is now between A and F.

Similar results have been obtained by Lewis Boss 9 from
a discussion of the proper motions of stars. The method,
which had previously been applied by Kapteyn to the
Bradley proper motions, depends on the following prin-
ciples. Let the proper motion be resolved into two
components, the parallactic motion v towards the solar
antapex and the cross proper motion T at right angles to
it. We can determine the mean parallax of the stars of
any type from the mean parallactic motion, by the aid of
the known speed of the solar motion. By means of this
mean parallax, the mean value of T, regardless of sign, can
be converted into linear measure. These linear cross-
motions are exactly comparable with the radial motions
that have just been discussed. Like them they are free
from the effects of solar motion, but are not corrected for
the star-streaming. Boss's results, which depend on the
excellent data of his catalogue, are as follows :

TABLE 24.

Min Velocities of Stars (Boss).

Type.*

Cross Linear Motion.

Weight
(No. of Stars).

B

6-3 " i 49b'""" j

A

10-2

1647

F

16-2

656

G

18-6

444

K

15-1

1227

M

17-3

222

* In Boss's classification, B includes Oe 6 to B 5 ; A includes B 8 to A 4 ;
F includes A 5 to F 9.

VIII

SPECTRAL TYPE 159

The close agreement with the quite independent
evidence of the radial velocities is very satisfactory.
Boss's results depend on the assumption that the solar
motion is the same for all types, which is open to some
doubt. As regards the irregularity of the progression F,
G, K, there is little doubt that his method of excluding
stars of excessive proper motion leads to too small a value
of the parallactic motion as compared with the cross
motion ; and this is especially the case for Types F and G,
which contain by far the largest proportion of great
proper motions. The linear motions deduced by him for
these two types should accordingly be diminished.

The facts here brought before us direct attention to the
very deep-lying question, How do the individual motions
of the stars arise ? It appears that as the life-history of a
star is traced backwards, its velocity is found to be smaller
and smaller. In the Orion stage it is only a third of what
it will ultimately become. Must we infer that a star is
born without motion and gradually acquires one ? I
believe this is the right conclusion, although there is
more than one loophole of escape, which deserves con-
sideration.

J. Halm 10 has suggested that equipartition of energy
holds in the stellar system ; according to his view the
Orion stars move slowly, not because they are young, but
because they are massive. If the stars were all formed
about the same epoch, the large stars might be expected
to take longer to pass through their stages of development
than the smaller stars, so that at the present time the
more massive the star the earlier would be its spectral
type. The main direct evidence as to the masses of the
stars is found in a discussion of the spectroscopic binaries of
which the orbits have been investigated. In cases where
both components are bright enough to show their spectra
the quantity (m^ + m*) sin 3 i can be found ; here / is the
unknown inclination of the orbit to the plane of the sky.

160 STELLAR MOVEMENTS CHAP.

There are available for discussion seven binaries of Type B
and nine of Types A G. Assuming that the mean
value of sin 3 1 will be the same for both groups, it is
found that

average mass of Type B binaries _ g.g
average mass of other types

When the spectrum of one component only can be
observed the quantity

' J?!_Y.in,-

m/

can be found. There are seventy-three suitable orbits of
this kind known. These give

average mass of Type B _ g.g
average mass of other types

These results indicate that the B stars are considerably
more massive than the other types ; and the ratio
actually agrees with that demanded by the law of equi-
partition of energy, viz., the average mass is inversely
proportional to the square of the average velocity. But the
main argument for equipartition has been a theoretical
one, depending on a supposed analogy between the
behaviour of stars and the molecules of a gas. This
subject will be considered in Chapter XII. ; the evidence
there given seems convincing that the analogy of the
stellar system with a gas system does not hold good ;
and equipartition, if it exists, cannot be explained in
this way.

It seems certain that the motion of a star has not
during the period of its existence been appreciably
disturbed by the chance passage of neighbouring stars.
This doctrine of non-interference leads to the conception
that each star describes a smooth orbit (not necessarily
closed) under the central attraction of the whole stellar
system. Such a star will wander sometimes near the

vin SPECTRAL TYPE 161

centre, sometimes at a remote distance, transforming
potential into kinetic energy and vice versd. The
nearer it is to the centre the greater will be its speed.
Consequently in the stellar system the average speed
may be expected to diminish from the centre outwards.
Ihis conclusion depends on the view that most of the
stars are continually approaching and receding from the
centre ; if the majority were describing circular orbits
the speed would actually increase from the centre outwards.
But accepting it as a possible and fairly likely condition,
it offers another explanation of the association between
velocity and spectral type. Suppose the Orion stars
move slowly, not because they are young, but because
they are very distant. The order of spectral type is
(or was until recently) believed to be the order of
luminosity and, consequently, for stars down to a limit-
ing magnitude, the order of mean distance. Thus it
may be that we are using the spectral classification as a
distance classification, and determining a relation between
distance and speed.

This explanation was formerly put forward in a tenta-
tive manner by the writer, 11 but it is given here only
that it may be disproved. To test it, the radial motions
of the stars of Type A were taken and grouped according
to the magnitude of the proper motion. This grouping
is a rough division according to distance, since the larger
proper motions usually indicate the nearer stars.

Centennial
Proper Motion.

Radiaf Velocity. | ^o. of Star,

!

"

km. per sec.

>20

10-1

19

1220

8-8 29

812

12-4

38

48

11-6

61

04

11-1

65

162

STELLAR MOVEMENTS

CHAP.

There is here no sign of a decreasing speed with
increasing distance. It is clear that distance cannot be
the determining factor. 12

We are thus thrown back on the original and straight-
forward conclusion that the phenomenon is a genuine
correlation between speed and spectral type, independent
of either mass or distance.

We have up to now been discussing the relation between
spectral type and the individual stellar motions ; it
remains to be considered whether the systematic
motions vary from one type to another. It was found in
Chapter V. that the declination of the solar apex
depended on the type of stars chosen, being more
northerly for the later types. It is not clear whether the
speed of the solar motion is appreciably different. The
following results are given by Campbell, 13 but the amount
of data is scarcely sufficient to allow of much weight being
attached to them :

Type.

Solar Velocity.

No. of Stars.

B
A
F
G
K
M

km. per sec.
20-2
15-3
15-8
16-0
21-2
22-6

225
212
185
128
382
73

According to the two-drift hypothesis the solar or
parallactic motion is merely the mean of two partially
opposing drift-motions, and for a fuller understanding of
these changes, or possible changes, of the solar motion
reference must be made to the drifts. It has been found
by many independent researches that the star-streaming
tendency is scarcely shown in the Type B stars, that

vin SPECTRAL TYPE 163

it is most strongly shown in Type A, and it becomes less
marked in succeeding types, though still quite prominent in
Type K. The sudden development of the star-streaming
in its full intensity in passing from Type B to Type A is
a curious phenomenon, but the evidence for it is over-
whelming.

The question has been studied quantitatively by H. A.
Weersma 14 from the data of Boss's Catalogue. If Fis the
velocity of one drift relative to the other, Q the mean
individual speed of the stars, he finds,

For Type A ZL - 2'29 0'19

For Types K and M . = Q'98 O'll

Again if P be the solar motion relative to the mean of the

stars,

For Type A .... ?l 1'08 0'08

For Types K and M . ?* = 0'62 0'04
Q 2

It was assumed in the investigation that the proportion
in which the stars are divided between the two drifts is
the same for Type A as for K and M, viz., 3:2. It is by
no means certain that this is correct.

These differences between the quantities F/ft, P/Q for
the two groups are largely accounted for by the differences
in ft that have already been discussed ; but it would
appear that to reconcile the results we must have also P l
different from P. 2 or else V l different from F 2 . Having
regard to the probable errors the evidence for this is
rather slight. If, for example, we put fi. 2 : HI =1*8, a
value which appears to represent the results derived from
the radial velocities (p. 158), then

v l -. V 2 = 1-30

P, : P 2 = 0-97

which makes the solar motion about equal for the two

164 STELLAR MOVEMENTS CHAP.

types, and gives a real diminution of the star-stream
velocity, in passing from Type A onwards. With a some-
what smaller ratio ft 2 : n x we should obtain the same star-
stream velocity, but a smaller solar motion for Type A
than Types K and M, an equally likely explanation,
which, moreover, receives a little support from the direct
determinations of the solar motion already quoted.

The view favoured by Kapteyn 15 abandons the assump-
tion that the division between the two drifts is the same
for all types. Instead there is a continuous increase in the
proportion of Drift II. stars as the spectral type advances,
and at the same time a continuous change in the direction
of the stream motions. He considers that in the course of
time the stream -motions have slightly changed in such a
way that the oldest stars have deviated most and the
youngest least, but all in a higher or lower degree, from
the original direction and velocity.

The conspicuous relation between the luminosities of the
stars and their spectral types has already been touched
upon in discussing the nearest stars. Much further
information can be gained from investigations of the
general mass of the stars. In consequence of the fact that
we usually consider catalogues or selections of stars
limited by a certain apparent magnitude, the difference in
luminosity leads to a difference in the average distance of
the spectral classes. In saying, as we commonly do, that the
B stars are more remote than the A stars, we do not mean
that there is any difference in their real distribution in
space, but only that, when we consider stars limited
by a certain magnitude, the selection of B stars is
dispersed through a larger volume of space than the
selection of A stars.

We might hope to gain information as to the average ;
distances, and therefore as to luminosities of the spectral j
types, by comparing their degrees of concentration to the j

VIII

SPECTRAL TYPE

165

galactic plane. The general tendency of the stars to
crowd to the galactic plane is explained by the oblate
shape of the stellar system, so that we see through a
greater depth in some directions than in others. But,
clearly, if a class of stars is confined to a small sphere
in the centre of the stellar system, its distribution will
not in any way be affected by the shape of the boundary.
Thus we find that the stars with proper motions greater
than 10" per century show no galactic concentration ; they
are all comparatively near to us. The greater the average
distance of the stars, or the wider the volume of space
through which they can be seen, the more will the oblate
shape of the system affect them. The deficiency of stars
in the region of the galactic poles will be more and more
marked. Thus we may expect the amount of galactic
concentration to be a measure of the average distance of
the class.

From the Revised Harvard Photometry, E. C. Pickering 16
has determined the distribution of the stars down to a
limiting magnitude of 6*5, arranged according to
spectral type and galactic latitude. His results are given
in Table 25.

TABLE 25.
Distribution of the Stars brighter than 6 m- 5.

Mean

Zone.

Galactic B.

A.

F. G.

K.

M.

Latitude.

I.

o
+ 62-:^ 8

189

79 61

176

56

II.

-f-41'8 28

184

58 69

174

49

III.

+ 21-0 69

263

83 70

212

57

IV.

-f 9-2 206

323

96 99

266

77

V.

- 7'0 161

382

116 84

239

45

VI.

-22-2 158

276

117 100

247

69

VII.

-38-2 57

161

94 59

203

59

VIII.

-62-3 29

107

77 67

202

45

1 66 STELLAR MOVEMENTS CHAP.

The eight zones are of equal area, so that the numbers
show directly the relative density at different galactic
latitudes.

Pickering's division of the spectral types was as follows :
B = O-B8; A = B9- L \3; F = A4-F2; G = F5 - G ;
K = G5 K2; M = K5 N. The divisions are somewhat
different from those we have previously considered.

Taking the degree of concentration shown in these

o o

tables as a measure of average distance, we should arrange
the types in the order of decreasing distance and decreasing
luminosity, thus :

B, A, F^FG, K, M,

which is identical with the order of evolution usually
accepted.

This agrees well with the results of Chapter III. as to
the luminosity of the stars derived from parallax investiga-
tions. A general decrease in luminosity with advancing
type was there noted. Further, as the sequence B, A, F,
G, K, M is probably the order of decreasing temperature,
it is not surprising that the luminosity should decrease in
the same way.

Nevertheless, this order is undoubtedly wrong. It is
not difficult to measure the average distances of stars of the
spectral types by less hypothetical methods. The mean
parallactic motion in arc is proportional to the mean
parallax, for the true linear parallactic motion is, at least
approximately, the same for all the spectral classes. Or,
again, by comparing the mean cross-proper motion (at
right angles to the parallactic motion) in arc with the
mean cross-motion in linear measure (Table 23), an
independent determination of the mean distance is
obtained. Five invest i-j.-u ions on these lines may be
cited. 17

VIII

SPECTRAL TYPE

167

TABLE 26.
Mean Distances of the Spectral Types.

(a) L. Boss.

(6) J

C. Kapteyn.

Type.

Parallactie
Motion.

No. of
Stars.

Type.

Mean
Parallax.

No. of
Stars.

Oe5 B5

2-73

490

B

//
0-0068

440

B8 A4

4-08

1647

A

0-0098

1088

A5 F9

4-99

656

F, G, K

0-0224

1036

G

3-12

444

M

0-0111

101

K

4-03

1227

M

3-29

222

(c) W.

W. Campbell.

(d) H. S. Jones.

Type.

Mean
Parallax.

No. of
Stars.

Type.

Mean
Parallax.

No. of
Stars.

BO-B5

0-0061

312

BO-B5

0-0031

11

B8, B9

0-0129

90

B8 A4

0-0058

188

A

00166

172

A5 F9

0-0110

187

F

0-0354

180

GO G5

0-0076

141

G

0-0223

118

G6 M

0-0056

140

K

0-0146

.346

M

0-0106

71

(e) K. Schwarzschild.

AP Ty P l mate Colour Index.

Parallactie
Motion.

No. of Stars.

m.

B

-0-65

3-5

64

A

-0-35

2-9

332

F

-0-05

8-9

277

G

+ 0-25 20-8

150

+ 0'55 8'6

126

K +0-85

7-6

277

+ 1-15

4-9

199

+ 1-45 4-0

184

M +175 4-6

71

The parallactic motion (centennial) is 410 times the parallax.

i68 STELLAR MOVEMENTS CHAP.

(a) L. Boss's results, based on the proper motions of
stars brighter than 6 m> in his Catalogue, refer to much
the same stars as those used in Pickering's discussion of
galactic distribution. Unfortunately Boss rejected all
proper motions greater than 20" per century ; this has
not only made his values systematically too small, but it
has had a disproportionately great etfect in the case of
Types F and G, which include the bulk of the stars with
excessively great motions. Accordingly the values for F and
G need to be greatly increased.

(b) Kapteyn's results depend on less accurate proper
motions than the preceding. To allow for differences in
the mean magnitudes of the different types, the values of
the mean parallax have been corrected so as to correspond
to magnitude 5*0.

(c) Campbell's determination is based on the cross-
motions. It refers to stars somewhat brighter than the
other investigations, the mean magnitude being 4 m *3.

(d) Jones's determination depends on the parallactic
motions of stars between Dec. 4-73 and +90, of mean
magnitude 6 m '8. The difference of 2*5 magnitudes between
these stars and Campbell's accounts for the smaller paral-
laxes found.

(e) Schwarzschild's classification is primarily by colour-
index. The proper motions were taken from Boss's
Catalogue.

All the investigations agree in showing that the mean
parallax increases steadily from Type B to a point some-
where about F or G, and then decreases again to a small
value for Type M. The order of distance is thus altogether
different from the standard order B, A, F, G, K, M. In
particular it appears that the stars of Type M are more
distant than any other type except B.

How then is it that the M stars show practically no
galactic concentration, whereas the A stars are strongly
condensed ? Our previous explanation fails, because the

vin SPECTRAL TYPE 169

assumption that Type M is much less remote than Type A
is now shown to be false. It seems necessary to conclude
that the apparent differences in galactic distribution are
real : that the system of the A stars is very oblate, and
the system of Type M is almost globular.

This leads to the following theory. The stars are
formed mainly in the galactic plane. Type B, on account
of the low individual speeds and the short time elapsed
since birth, remains strongly condensed in the plane. In
succeeding stages the stars have had time to stray farther
from the galactic plane, and their higher velocities assist
in dispersing them from it. In the latest type, M, the
stars have become almost uniformly scattered, and very
little trace remains of their original plane. We shall see
reason in Chapter XII. to modify this hypothesis slightly.

It will be seen that, in regard to the relation of spectral
type both to speed and to galactic concentration, we have
been driven to adopt the straightforward interpretation of
the phenomena that occurs most naturally to anyone who
has not considered the subject deeply. The correlation is
exactly what it appears to be, and subtle suggestions as to
its being mixed up with other effects are found to fail in
the end. Yet I think we have been right in not jumping
to the obvious conclusion at once ; it was necessary to
examine, and for a time prefer, the alternative explanations,
which, though more complex in themselves, led to a simpler
conception too simple it now appears of the stellar
system.

An outstanding point of great difficulty remains, which
may most conveniently be illustrated by the stars of
Type M. We have been led to two opposed views as
to their luminosity. In the parallax investigations of
Chapter III. they were found to be the faintest of all the
types ; in the present statistical investigations they are
found to be the most luminous, except Type B. Our

i yo STELLAR MOVEMENTS CHAP.

conclusion from -the parallax investigations may be judged
to rest on rather slight though very consistent evidence ;
but other (less trustworthy) parallaxes confirm it, and
moreover the K stars show a similar discordance in the two
kinds of investigation. It may be admitted at once that
the parallax and statistical results relate to entirely
different selections of stars ; none of the extremely feeble
M and K stars of Tables 3 and 5 enter into the data of
our last discussion. Both results are probably right ; but
it is difficult to see how they are to be reconciled.

The leading contribution to this problem is the
hypothesis of " giant" and " dwarf" stars put forward by
E. Hertzsprung 18 and H. N. Russell. 19 They consider each
spectral type to have two divisions, which are not in reality
closely related. The one class consists of intensely luminous
stars and the other of feeble stars, with little or no
transition between the two classes. Assuming that the
feeble stars are much more abundant than the luminous
stars in any volume of space, the parallax investigations
will take hold of the dwarfs mainly, whilst the statistical
investigations, selecting by magnitude, will be concerned
with the giants. This will account for the different
luminosities ; for Type M will then denote two entirely
different classes in the two kinds of research.

Russell has supported this hypothesis by direct evidence
from the parallax determinations. By his kindness I am
permitted to reproduce his diagram (Fig. 19) of the abso-
lute luminosities of all stars for which the necessary data
could be obtained ; parallaxes have been used to which
we should hesitate to attach much weight ; but the
principal features of his diagram can scarcely be doubted.
Below each spectral type are shown dots which represent
on a vertical scale the absolute magnitudes (magnitude
at a distance of 10 parsecs) of individual stars of that type.
The large circles represent mean values for bright stars of
small proper motion and parallax.

VIII

SPECTRAL TYPE

171

The general configuration of the dots seems to corres-
pond to two lines thus, "\ . There would appear to be two

M

IP

o x

\

It

o

Fi<;. 19. Absolute Magnitudes of Stars (Russell).

series of stars, one very bright and of brightness almost
independent of the spectrum, and the other diminishing

1 72 STELLAR MOVEMENTS CHAP.

rapidly in brightness with increasing redness. The former
series, corresponding to the horizontal line, are the giants,
and the latter, corresponding to the oblique line, the dwarfs.
For Types B and A the giants and dwarfs practically
coalesce ; but the divergence increases to a very large
amount at Type M. It must be remarked that the
evidence of this diagram, convincing as it looks, does not
compel us to divide Types K and M into two distinct
classes. The stars of which the parallax and luminosity have
been measured are in most cases chosen for brightness or
for nearness (large proper motion). The two groups may
thus result from the double mode of selection, without
implying any real division in the intrinsic luminosities.

For example, if the absolute magnitudes M are distri-
buted according to the frequency law

the stars of absolute magnitude J/ and of apparent
magnitude greater than m are those within a sphere of
radius r given by

lo glo r = 0'2(m-3f),

the volume of this sphere is proportional to r 3 or to

1QO-6 (m-H),

and the frequency of an absolute magnitude M among
stars limited by the magnitude m is proportional to

e - k'*(M - Jf )2 + 1 -38 (m - M ).

This is an error distribution with the same dispersion as

before but ranged about the mean value M -- ^-.

/c~

Thus our two methods of selecting parallax stars would
give luminosities clustering about two separate magnitudes

M and M . If it is supposed that }/k increases
/c

with advancing type, the two diverging groups will be
explained. From Fig. 19 it appears that the two groups

vin SPECTRAL TYPE 173

of Type M differ by about eleven magnitudes. Setting
0-69/7r=ll, we have l/& = 4 m '0. For Type G the
difference is six magnitudes, and l/k = 2'9.

Russell has shown that the error-law, assumed for the
absolute magnitudes, is confirmed by the observations ;
but the modulus is smaller than that calculated, viz.,
l/k=l m '6 (corresponding to a probable deviation O m '75).
This result may be considered to refer to the mean of all
spectral types.

Whilst the evidence from the directly determined
luminosities is thus scarcely conclusive, there are several
other indications which point to a real existence of the two
series. Perhaps the strongest argument is a theoretical
one. According to the well-known theories of Lane and
Ritter, as a star contracts from a highly diffused state,
its temperature rises until a certain concentration is
reached, after which the loss of heat by radiation is
greater than the gain by conversion of gravitational
energy into heat, and the star begins to cool again.
Recent discoveries of a new supply of energy in radio-
active processes involve some modification of these
theories ; but probably the general result of a tempera-
ture increasing to a maximum, and then diminishing, may
be accepted. Now, if the spectrum of a star depends chiefly
on its effective temperature, it may well be that the Draper
classification groups together stars of the same temperature
regardless of whether they are in the ascending or des-
cending state. The former would be highly diffused
bodies, and the latter concentrated. The surface bright-
ness, which depends on the temperature, being the same,
the ascending stars with large superficial area would give
a great deal more total light than the denser descending-
stars. If then the stars have all much the same mass
a conclusion supported by such evidence as is available
we shall have two groups in each type, one of low density
and intense total luminosity and one of high density and

STELLAR MOVEMENTS

CHAP.

low luminosity. These two groups will coalesce for the
B stars, which mark the maximum temperature reached,
and fall wider apart the lower the temperature, just as in
the diagram. Moreover, on the ascending side the
increasing temperature and diminishing superficial area
will oppose one another in their effect on the luminosity,
so that the change of brightness from type to type will be
small. On the descending side the decreasing surface and
decreasing surface-brightness will both lead to a rapid
change of luminosity from type to type.

The determinations of density of the visual and spectro-
scopic binaries (p. 24) favour the view that some of the
later type stars are in a very diffused condition, and that
others are very condensed. Table 27, due to H. Shapley, 20
contains the determinations of density of eclipsing systems.
Unfortunately, these are mainly early type stars, but the
tendency to divide into two groups is well illustrated even
in F and G.

TABLE 27.
Stellar Densities (Shapley).

Density
( Water = 1).

B.

A.

F.

G.

K.

>i-oo

1

1-00 -0-50

1

0-50 0-20

1

10

6

1

0-20 -0-10

4

12

1

1

0-10 0-05

3

17

0-05 -0 02

2

8

0-02 0-01

3

1

1

o-oi o-ooi

2

o-ooi o-oooi

2

1

< o-oooi

1

1

It is well-known that Lockyer's classification differen-
tiates the stars into series of ascending and descending
temperatures, but according to Russell the giant and
dwarf stars do not correspond to Lockyer's criteria.

VIII

SPECTRAL TYPE

With regard to the possibility of distinguishing two
series by slight differences in their spectra an interesting
contribution has been made by Hertzsprung. In Miss
Maury's classification certain stars are discriminated as
having what is known as the c character, marked by the
very sharply defined appearance of the absorption lines.
These stars (which are rather few in number) are found to
have much smaller proper motions than the corresponding
stars of the same spectral type without the c characteristic.
With one exception (v Ursae Majoris) the motions are
almost imperceptible, generally smaller even than those of
the early Orion type. If, in order to allow for differing
magnitudes, the proper motions are all multiplied so as to
represent what would be the apparent motion of the star at
a distance at which it would appear of zero magnitude, the
following (condensed from Her tzsp rung's table 21 ) shows the
results for stars with and without the c characteristic :

TABLE 28.
Stars with the c Characteristic.

*" MoE.

Normal
P.M.

c-Star.

Proper
Motion.

Normal
P.M.

o. 2 Can. Maj. . . 0'03

0-20

v Persei ....

0-06

n
1-20

67 Ophiu. ... 0-08

0-20

a Persei ....

0'09

2-22

Rigel O'OO

0-41

8 Can. Maj. . .

0-02

3-25

fi Sagittarii . . 0*03

0-41

p Cassiop. . . .

0-07

3-25

2 Camelop. . 0'05

0-52

y Cygni ....

o-oi

3-25

rj Leonis ... 0'03

0-39

Polaris ....

0-11

3-25

a Cygni .... O'Ol

0-44

T) Aquilae . . .

0-07

0-48

22 Androm. . . 0'09

072

a Aquarii . . .

0-07

0-48

a Leporis ... 0'02

0-57

10 Camelop. . .

0-08

0-48

TT Sagittarii . . 0*14

0-57

d Cephei . .

0-08

0-48

v Ursa; Maj. . . 1'96

0-57

C Geminorum .

0-02

0-48

Auriga? ... 0'07

1-20

(The proper motions are reduced to zero magnitude as standard. )

The second column gives the proper motion of the c-
stcir, the third column the average proper motion of the

176 STELLAR MOVEMENTS CHAP.

remaining stars of the same type in Miss Maury's classifi-
cation, in each case reduced to zero magnitude. The first
five stars are of Type BO to B9, the next six are from A to
F and the remainder are in sub-divisions of types F and
G. There are no K or M stars in the table.

It is clear that these stars with ^-characteristic must be
very remote and therefore (with the exception of v Ursae
Maj.) belong to the class of " giants." It looks as though
a beginning has been made in the direct discrimination of
the two groups by their spectra.

It will be seen that a very strong case has been made
out for the recognition of two divisions in the later
spectra] types, corresponding to widely different luminosi-
ties. Yet there is a great difficulty in accepting Russell's
theory in its entirety ; if it is right, it causes a revolution
in many of the results that have been generally accepted.
In particular we shall have to revise the supposed order of
evolution which has hitherto been assumed. Russell's
theory gives the complete order M x K : GI F x A! B A 2 F 2
G 2 K 2 M 2 , where the sutfix 1 refers to the giants and 2 to
the dwarfs. It is an essential part of his theory that the
dwarf stars of Types M and K are too faint to appear in
the statistical investigations of proper motions and radial
velocities ; that is to say, types M and K must be identi-
fied with M! and K 1} so far as these investigations are con-
cerned. Moreover, the values of the parallactic motion show
that the dwarfs of Types F and G play a predominant
part in such researches, for the average distance and
intrinsic brightness of these types is much less than for
Types B and A. Therefore, somewhere about Type G we
cross over, as it were, from the ascending branch to the
descending branch. Having regard to this, the order of
evolution becomes M K B A F G ; an order which applies
to all researches' depending on stars selected for magnitude.
From the astrophysical point of view the apparent breach
of continuity between K and B does not matter ; it is not

VIII

SPECTRAL TYPE 177

suggested that stars really pass from K to B at a jump,
but that the intermediate types are outnumbered in the
catalogues by stars indistinguishable from them in spectra
but in a later stage of evolution.

The new order M K B A F G upsets altogether the
regular progression of speed with type, and of galactic
concentration with type. We should have to suppose that
a star is born with a large velocity, that the speed
decreases almost to rest and then increases again. Even
if we do not insist on the predominance of dwarfs in F
and G (though by abandoning it we lose one of the advan-
tages of Russell's theory), and are content with simply
reversing the usual order of evolution, the difficulties are
great. We should have to amend our former hypothesis,
and suppose that the stars originate with large velocities in a
nearly spherical distribution, and afterwards become con-
centrated to the galactic plane, losing their velocities as
they do so. The explanation is not improved by being
turned upside down. And, further, it was shown in Chapter
III. that the feebly luminous stars the dwarfs of Types K
and M have extremely large velocities, so that in still later
stages the speeds must increase again, even beyond their
original amounts. The fact that both dwarfs and giants of
Types K and M have larger speeds than the other types
seems to imply a close connection between them, and it
is very unsatisfactory to place them at opposite ends
of the evolutionary scheme.*

There is another piece of evidence which lends strong
support to the generally accepted order of evolution.
Table 29 shows the periods of the spectroscopic binaries
arranged according to type ; it is due to Campbell."

* For Russell's reply to these criticisms, see The Observatory, April, 1914,
p. 165.

i 7 8

STELLAR MOVEMENTS

CHAP.

TABLE 29.
Periods of Spectroscopic Binaries (Campbell).

Period.

Type.

Total

Short.

O d 5*

5 d 10"

10 d -365 d

> 1 year

Long.

O and B

8

15

10

14

1

48

A

4

10

1

12

2

29

F

6

2

4

3

1

16

G

1

6

3

10

K

2

3

9 14

M

1

1 2

The columns headed " short " and " long " contain stars
the periods of which have not been determined.

The increase in period with advancing type is very
striking. It is also significant that so large a proportion
of the spectroscopic binaries are of early type, indicating
that the later types generally move too slowly to be de-
tected with the spectroscope. A rough classification by
E. G. Aitken of the more rapidly moving visual double
stars gave the following proportions :

Types O and B
A ,, F
G K

M , N

4 stars.
131

28
1

Thus components of Type B are rarely sufficiently far
apart to be seen separated ; and it would appear that in
Types M and N the separation is so great that they are
almost unrepresented on Aitken's list. If we believe that
double stars originate by fission, and that the components
separate farther and farther under the influence of tidal and
other forces,* as time advances, we cannot but regard
this result as a thorough vindication of the standard order
of the types. Moreover, in the case of the spectroscopic

* H. N. Russell has directed my attention to the fact that tidal forces are
competent to produce only a limited amount of separation.

vin SPECTRAL TYPE 179

binaries at least, we are dealing with stars selected for
brightness exactly as in the statistical investigations of
stellar motions just the selection for which the order is
challenged by Russell's hypothesis.

To sum up the present position, there is direct evidence
that in the later types the stars are of two grades of
luminosity, and that they are of two grades of density.
The former division might perhaps be due to two different
principles of selection of the stars being employed, though
that would leave unexplained why on the one principle
the late type stars are the faintest, and on the other
principle the brightest of the classes. If we associate the
divisions of luminosity with the divisions of density, as
the Lane-Ritter theory suggests, this upsets the usually
accepted order of evolution, so far as statistical investi-
gations are concerned, an order which has been indepen-
dently confirmed by studies of stellar velocities, galactic
distribution, and the periods of binary stars.*

We shall conclude this chapter with some general
remarks on the Orion type and fourth type stars, both of
which present some interesting features.

The position of the Orion or Type B stars is very
remarkable. Neither from their proper motions nor from
their radial velocities do they show any tendency to
partake of the motions of the two star-streams. If we had
to class them with one of the drifts by their motions, we
should naturally assign them to Drift I., but that is only
because the Drift I. motion approximates more nearly to
the parallactic motion than Drift II. does. Actually, such
systematic motion as there is seems to be purely paral-
lactic and due to the motion of the Sun in space. We
now know that this peculiarity of being at rest in space,
except for small individual motions, is shared by other

* Elsewhere in this book the point of view is always that of the older
theory not Russell's theory unless expressly stated.

N 2

1 8o STELLAR MOVEMENTS CHAP.

stars not belonging to this type. As we have seen, the
dissection of the stellar motions into two streams leaves
over a certain excess of stars (including both A and K
types) moving towards the solar antapex, and presumably
therefore at rest when the parallactic motion is removed.

A notable feature of the Type B stars is their tendency
to aggregate into moving clusters. " Moving Clusters " is
perhaps rather a misnomer, for the motion is usually very
small ; but they are groups apparently analogous to the
Hyades cluster. The great Scorpius-Centaurus cluster, the
constellation Orion, the Pleiades and the Perseus cluster
between them account for a considerable proportion of the
known stars of this type. The distance of these groups
appears to be from about seventy to one hundred parsecs,
except perhaps in Orion, which may be more distant. The
remaining stars of the type have generally considerably
smaller proper motions than these cluster stars, and are
judged to be more distant still. This was well shown in
Fig. 4 in which the non- cluster stars, forming the group
of crosses near the origin, have barely appreciable proper
motions. Lewis Boss 23 finds from his discussion of proper
motions that a space round the Sun having a radius of
seventy parsecs (corresponding to a parallax //g 015) is
" almost wholly devoid of these stars." Such a space
would, according to the conclusions of Chapter III., contain
at least 70,000 stars of other types. There seems no
reason to believe that the part of space around our Sun
is unusually bare of Type B stars ; it is rather to be
supposed that their general distribution is extremely rare,
but that owing to brightness they are visible at say ten
times the distance of an ordinary star and therefore
through a space a thousand times as great. Their distri-
bution, though rare on the average, is irregular, and in the
moving clusters there must be many comparatively crowded
together.

The assumption that the proper motion gives a measure

VIII

SPECTRAL TYPE

181

of the distance of these stars is more than usually justi-
fiable in this case. Owing to their small individual
motions, and to the absence of star-streaming, the whole
motion cannot generally differ much from the parallactic
motion. Omitting the divisions B8 and B9 which are
probably more closely allied with Type A, there is only one
known case of a large proper motion, that of the star a
Gruis with a motion of 20"'2. Its parallax (determined
by Gill with great accuracy) is only 0"'024, so that its
linear velocity must be remarkably great for its class. No
other stars from BO to B7 have centennial motions so great
as 10". Of late Type B stars (B8 and B9) the motions of
Regulus, 25", and of 13 Tauri. 18" per century, are unusually
large.

The fourth type stars (Type N) are for the most part
too faint to come into the general discussions of distribution
and motions. In Pickering's table of the galactic distribu-
tion of the types, the few that were included were classed
with the M stars. Happily there are too few of them to
affect the figures appreciably, for they present a very
strong contrast to Type M in their distribution. It has
been shown by T. E. Espin 24 and J. A. Parkhurst 25 that
they are strongly concentrated to the galactic plane, as the
following table shows :

TABLE 30.
Galactic Distribution of Type N.

Number of N -Stars.

Relative Density. Density

Galactic

of Durch-

Latitude.

musterung

Espin.

Parkhurst.

Espin.

Parkhurst.

Stars.

05 1
5-10 /

123

( 92 1
1 46 1

11-4

( 18-3
\ 9-2

27
2-6

1020 43

58

4-0

6-0 2-1

20-30 27 17

3-0

1-9 1-5

>30 31 29

1-0

i-o i-o

1 82 STELLAR MOVEMENTS CHAP.

The concentration is even a little stronger than for the
Orion stars, but allowance should be made for the fact
that we are using a much fainter limit of magnitude
for these than for Pickering's table. The fainter the
magnitude and the greater the distance, the stronger
should be the apparent concentration to the galactic plane,
as is indeed borne out by observation. Making allowance
for this, Type N may probably be placed between B and A
in the order of galactic condensation.

From 120 stars of this type of average magnitude 8 m< 2,
J. C. Kapteyn 26 has determined the parallactic motion ; this
is found to be 0"'30 per century with a probable error of
practically the same amount. The corresponding parallax
would be 0"'0007itO"-0007. For the Orion stars the paral-
lax found by the same method was 0"-00680"*0004 for
magnitude 5'0 (agreeing with other determinations already
quoted). These N stars are thus many times more remote
than the Orion stars. Their luminosity, however, may be
slightly less, the difference 3 m< 2 in apparent magnitude
counterbalancing the greater distance.

Hale, Ellerman, and Parkhurst 27 have pointed out that
the fourth type stars possibly have certain features in
common with the Wolf-Rayet type. But they saw no
reason to believe that any important organic relationship
<>! meets the two types.

REFERENCES. CHAPTER VIII.

1. Monck, Astronomy and A*ti-<>p/ii/>c8, Vols. 11 and 12.

2. Kapteyn, Astr. Nach., No. 3487.

3. Dyson, Proc. Roy. Soc. Edinburgh, Vol. 29, p. 378.

4. Frost and Adams, Yerkes Decennial Publications, Vol. 2, p. 143.

5. Eddington, Nature, Vol. 76, p. 250 ; Dyson, loc. cit., pp. 389, 390.

6. Kapteyn, Astrophysical Journal, Vol. 31, p. 258.

7. Campbell, Lick Bulletin, No. 196.

8. Innes, The O/^-,-,-,//,,,-,/, v..l. :',>, p. 270.

9. Boss, Astron. Jcmrn., Nos. 623-4, p. 198.

10. Halm, Monthly Notice*, Vol. 71. p. ;:;i.

11. Eddington, /*///. Auoc. /.v,,,,,-/, 1911, 'p. 259.

12. See also Kapteyn, Proc. Amnh'nlum JW., 1911, pp. 528, 911.

vin SPECTRAL TYPE 183

13. Campbell, Lick Bulletin, No. 211.

14. Weersma, Astrophysical Journal, Vol. 34, p. 32 -V

15. Kapteyn. Pr<><-. Amsterdam Acad., 1911, p. .VJ4.

16. Pickering, Harvard A;mals, Vol. 64, p. 144.

17. L. Boss, A.ttrnn. Journ., Nos. 623-4 ; Kapteyn, Astrophysical Journal,
Vol. 32, p. 95; Campbell, Lick Bulletin, No. 190, p. 132; Jones, Monthly
Notices, Vol. 74, p. 168 ; Schwarzschild, Gottingen Aktwometrie Teil B.,
p. 37.

18. Hertzsprung, Zeit. fur. Wiss. Phot., Vol. 3, p. 429 ; Vol. 5, p. 86 ;
Axtr. Nach., No. 4296.

19. Russell, The Observatory, Vol. 36, p. 324 ; Vol. 37, p. 165.

20. Shapley, Astrophysical Journal, Vol. 38, p. 158.

21. Hertzsprung, Zeit. fur. Wins. Phot., Vol. 5, p. 86.

22. Campbell, Stellar Motions, p. 260.

23. Boss, Astron. Journ. t Nos. 623-4.

24. Espin, Astrophysical Journal, Vol. 10, p. 169.

25. Parkhurst, Yerkes Decennial Publications, Vol. 2, p. 127.

26. Kapteyn, Astrophysical Journal, Vol. 32, p. 91.

27. Hale, Ellerman, and Parkhurst, Yerkes Decennial Publications,
Vol. 2, p. 253.

CHAPTER IX

COUNTS OF STARS

IN the investigations described in the four preceding
chapters we have generally been confined to stars brighter
than the seventh magnitude. Occasional excursions have
been made beyond that limit, and stars down to the ninth
or tenth magnitudes have made some contribution to our
knowledge ; further than this we have been unable to go.
Beyond the tenth magnitude there is an ever-increasing
multitude of stars, which hold their secret securely. We
know nothing of their parallaxes, nothing of their spectra,
nothing of their motions. There is only one thing we can
do count them. Carefully compiled statistics of the
number of stars down to definite limits of faintness can
still yield information which is of value for our purpose.

The fundamental theorem relating to these statistics is
as follows :

In a stellar system of unlimited extent in which the
stars are scattered uniformly, the ratio of the number of
stars of any magnitude to the number of stars one
magnitude brighter is 3 '98.

The ratio alluded to is usually called the star-ratio. If
in any direction it is found that the star-ratio falls below
the theoretical value 3*98, this shows that we have
penetrated so far as to detect a thinning out in the density
of distribution of the stars. It is assumed that the
absorption of light in space is negligible.

184

CH. ix COUNTS OF STARS 185

The number 3 '98 is equivalent to (2*512)* and the
formula may be written :

star -ratio for one magnitude = (light-ratio for one magnitude)?.

In this form the theorem becomes fairly evident. A light
ratio of 2*512 involves a distance-ratio of (2*512)^, and a
volume-ratio (2*512)*. That is, for every small volume of
space S at a distance D, there will be a corresponding
volume (2'512)*ata distance (2'512)iD, such that the
distribution of apparent magnitudes of the stars in the two
volumes will correspond except for a difference of one
magnitude due to the distance factor. But there will be
(2 '5 12)* times as many stars in the second volume as in
the first. Hence, a drop of one magnitude multiplies the
number of stars by the factor 3 '9 8 through the whole
range. The fact that the stars are of varying degrees of
intrinsic brightness is taken into account in this
argument.

The thinning-out of the stars at great distances from the
Sun manifests itself in a gradually-decreasing value of the
star-ratio for successively fainter magnitudes. It is of
great importance to have accurate knowledge of the rate
at which the star-ratio diminishes, and particularly of the
way in which it is related to galactic latitude. It may be
hoped that the information may lead to a more precise
knowledge of the flattening of the stellar system towards
the plane of the Milky Way.

The value of any compilation of star-counts will depend
mainly on the accuracy with which the magnitudes, to
which they refer, have been determined. In modern
researches, the standardising of the counts by a special
photometric investigation is a sine qua non. But it is
only recently that this refinement has come within
practical possibilities ; and much statistical matter that
has been used up to now depends on the ingenious adapta-
tion and correction of data, which were initially rather un-

1 86 STELLAR MOVEMENTS CHAP.

suitable. It must be admitted that these early researches
accomplished their end in the main ; and that they have not
only prepared the way for more satisfactory determinations,
but also have taught us much with regard to stellar distri-
bution that has a lasting value. But having now available
sufficient statistics based on sound magnitude-standards,
we shall not need to recur to the pioneer discussions,
except where discrepancies of particular interest arise.

An investigation of the number of stars of each magni-
tude by S. Chapman and P. J. Melotte, 1 published in 1914,
contains by far the most comprehensive treatment of this
problem, and we shall attach the greatest weight to it.
The magnitudes are photographic magnitudes based on the
Harvard Standard North Polar Sequence. The general
accuracy of the Harvard magnitude-scale has been
confirmed by investigations made at Mount Wilson and
Greenwich ; and for our present purposes it is believed to
be accurate enough ; it is possible, however, that the
corrections may not be altogether negligible in future
more elaborate discussions. The statistics given by
Chapman and Melotte extend from magnitude 2 m '0 to
17 ra *0. This huge range (representing a light-ratio of
1,000,000 to 1) is filled in almost continuously by data
derived from five separate investigations. As each
investigation is particularly strong near the middle
point of its range, there are five well-determined points.
These alone should suffice to give a correct idea of the
course of the star-numbers throughout the fifteen magni-
tude intervals, even without the weaker results, which
bridge the gaps.

The five sources of data are as follows :

(1) Magnitude 12 to 17*5. Counts on the Franklin-
Adams chart of the sky. These contain the results from
750 areas scattered over the northern hemisphere, each
containing from 60 to 90 stars in all. This represents
only a portion of the counting of the Franklin-Adams

IX

COUNTS OF STARS 187

chart carried out at Greenwich; but for the other areas
the comparison with the standard sequence has not yet
been effected, and, accordingly, the results are not used.

(2) Magnitude 9 to 12*5. Counts of stars in the
Greenwich Astrographic Catalogue (Dec. +64 to +90).
For 19f) plates the formulae for reducing the measured
diameter, published in the catalogue, to magnitude had
been determined by rigorous comparison with the standard
sequence, and these results were used for the present
investigation.

(3) Magnitude 6*5 to 9. Counts of stars in the Green-
wich Catalogue of Photographic Magnitudes of Stars
brighter than 9 m '0 between Declination + 75 and the Pole.
These magnitude-determinations were made from plates
specially taken for the purpose with a portrait-lens, the
Astrographic plates being unsuitable for stars of this
brightness.

(4) Magnitude 5 to 7 '5. Counts of stars in Schwarz-
schild's Gottingen Actinometry for Dec. to + 20. A
small correction (O m * 13) was required to reduce from the
Gottingen to the Harvard scale of magnitudes.

(5) Magnitude 2*0 to 4*5. Counts of stars in the
Harvard catalogue of photographic magnitudes of bright
stars (Harvard Annals, Vol. 71, Pt. I.). This is the least
satisfactory part of the data, for the magnitudes were not
determined photographically, but were found from the
visual magnitudes by applying the colour index corre-
sponding to the known spectral type of each star. The
magnitudes were published before the standard sequence
appeared, and it is not clear how far they conform to
that scale.

As the principal feature in the apparent distribution of
the stars is the variation with galactic latitude, the data
have been arranged in eight galactic belts. The first seven
belts are from 10, 1020, .... 60 70, and
belt VIII is from 70 90 galactic latitude (North or

i88

STELLAR MOVEMENTS

CHAP.

South). The sources of data (l). (4), (5) cover the whole
range, but (2) and (3) are confined to the belts I V

Magnitude

5 6 7 8 9 10 11 12 13 14 15 16 17
I I I I I I I I I III

- log B

m

I I I I I
12 13 14 15 16

I I I I I I
5 6 7 8 9 10 11

Magnitude

Fio. 20. Number of stars brighter than each magnitude for eight zones.
(Chapman and Melotte).

and II V respectively ; thus the information is not so
complete for the three highest zones as for the remainder.

IX

COUNTS OF STARS

189

If B m is the number of stars per square degree brighter
than the magnitude w, the counts are most conveniently
exhibited by plotting log B m against ra. This is done for
each of the eight belts in Fig. 20, and smooth curves have
been drawn to represent the results. In order to prevent
overlapping of the eight curves, they have been displaced
successively through an amount O'o in the vertical direc-
tion. The lowest curve corresponds to belt I. The
alternation of dots and crosses serves to differentiate the
four sources of data. The data (5) are not shown.

It is the central part of the data from each source that
is best determined, and the outer parts may be expected
to run off the curve a little. The general agreement of
the separate sets of data is very satisfactory.

The values of log B m for each magnitude, as read from
the curves, are given in Table 31.

TABLE 31.
Log B m for each Magnitude (Chapman and Melotte).

Zone.

I.

II.

III.

IV.

V. VI. VII. VIII.

Whole
Sky.

G'ilElCtlC

Latitude.

10

10' 20"

20' 30

30 40'

40 50 50 60 60 70 70* 90

90

Magnitude

ra.

5-0

2-435

2-360

2-170

5-055

2-040 2-030 2-105 2'120

2-223

6-0

1-010

2-950

2-800

2-700

2-655 2-610 2-660 2'680

2-819

7-0

1-555

1-500

] -385

1-295

1-215 1-170 1-180 1-200

1-377

8-0

0-065

0-015

1-930

1-830

1-730 1-685 1 670 1*670

1-895

9-0

0-545

0-490

0-420

0-320

0-200 0-165 0-125 0'115

0374

10-0

0990

0-935

0-880

0-770

0-640 0-605 0-550 0'520

0-819

11-0

1-405

1 -345

1-300

1-180

1-030 1-010 0-940 0-900

1-229

12-0

1-790

1 7'2.->

1-680

1 -.54.5

1-385 1-385 1-305 1-2.55

1-605

13-0

2-150

2-075

2-020

1-880

1-715 1-730 1-645 1*585

1-951

14-0

2-485

2-405

2-340

2-185

2-020 2-045 1965 1-890

2-268

15-0

2-800

2-715

2-630

2-465

2-300 2-33.5 2-265 2190

2-575

16-0

3-095

3005

2-900

2-720

2-565 2600 2 -.540 2 "470

2-855

17-0

3-380

3-285

3-15.5

2-965

2-815 2-850 2-810 2'74.5

3-12.5

It is of interest to find the total number of stars in
the sky, so far as can be deduced from the samples dis-
cussed. The results are as follows :

190

STELLAR MOVEMENTS

CHAP

TABLE 32.

Number of Stars in the Sky brighter than a given Magnitude
(Chapman and Melotte).

Limiting Number of

Limiting

Number of

Magnitude. Stars.

Magnitude.

Stars.

m.

m.

5-0

689 12-0 1,659,000

6-0

2,715 13'0 3,682,000

7-0

9,810 14-0 7,646,000

8-0

32,360

15-0 15,470,000

9-0

97,400

16'0 29,510,000

10-0

271,800

17-0 54,900,000

11-0

698,000

The curves in Fig. 20 are approximately parabolic arcs,
and the results can be expressed with satisfactory accuracy
by empirical formulae of the type,

(1)

But it is more convenient to use m 11 instead of m, since
the zero of magnitude is outside the range we are con-
sidering. The formulae for the eight zones are as
follows :

TABLE 33.
Number of Stars brighter than a gicen Magnitude.

Zone I. log 10 B w = 1'404

+ 0-409 (m-11) -0-0139 (m-11) 2

II. . .

= 1-345

+ 0-407

-0-0147

III. . .

= 1-300

+ 0-411

-0-0193

IV. . .

= 1-177

+ 0-403

-0-0186

V. . .

= 1-029

+ 0-391

-0-0168

VI. . .

= 1-008

+ 0-399

-0-0160

VII. . .

= 0-941

+ 0-389

-0-0135

VIII.

= 0-901

+ 0-380

-0-0130

The formulae make it clear that the main part of the
variation with galactic latitude consists in a change of the
constant term. The coefficient of (m 11) is nearly
stationary, and that of (m II) 2 shows no systematic
progression with latitude. The ratio of the star-density

ix COUNTS OF STARS 191

near the galactic pole to that near the galactic plane
is practically the same for all magnitudes. Or, taking
another point of view, the rate of increase in the number
of stars with advancing magnitude follows the same law
for all latitudes.

This conclusion is, of course, only approximate. We see
from Table 31 that the ratio of the star-density in Zone 1
to that in Zone VIII is

For magnitude 6 m '0, 2'1 : 1
17 m '0, 4-3 : 1

from which the conclusion may be drawn that the rate
of increase in the number of stars is appreciably greater
near the galactic plane than away from it. But these
differences are very slight compared with those found in
some former investigations, which have hitherto found
wide acceptance. In particular the celebrated star-gauges
of the Herschels, which have dominated our views as to the
distribution of faint stars for nearly a century, gave widely
different results. Prior to Chapman and Melotte's work
the most extensive discussion of magnitude statistics was
that of J. C. Kapteyn 2 (1908), which gave a very different
idea of the effect of galactic latitude on the star-density.
From Kapteyn's table it appears that the ratio of the star-
density in Zone I to that in Zone VIII should be

For magnitude 6 m '0, 2'2 : 1
17 m '0, 45 : 1.

It is difficult to explain the huge difference between
Kapteyn's result for the distribution of the faint stars, and
that of Chapman and Melotte. The former research was
somewhat provisional in character, an interim result to
be used until data on a more uniform plan could be
obtained ; indeed the complete charting of the heavens by
J. Franklin-Adams was largely inspired by the influence of
Kapteyn, jointly with Sir David Gill, for the purpose of
obtaining more satisfactory statistics. Yet the great

192 STELLAR MOVEMENTS CHAP.

divergence between the old and the new is surprising, and
it may be well to attempt to trace the precise source from
which it arises.

In the first place it must be remarked that Kapteyn's
figures for 17'"'0 are an extrapolation ; his data did
not go beyond 14 IU *0. If then we compare the results
for 14 m< 0, we have to account for a discordance-
Ratio Star-density, Zone I to Zone VIII Kapteyn II 1 5 : 1
,, ,, ,, ,, Chapman and Melotte 3 '9 : 1.

Kapteyn made use of seven main sources of information.
Of these the first four (including counts on the plates of
the Cape Photographic Durchmusterung) relate to stars
brighter than 9 m> 25 ; these show no important divergence
in galactic distribution from the new fi (Hires. For the

O o

fainter stars the main reliance was placed on Sir John
Herschel's star-gauges, 3 i.e., counts of stars visible with his
18-inch reflector in a field of definite area. Although these
are confined to the southern hemisphere, they are more
evenly distributed and more typical of normal parts of the
heavens than those of Sir William Herschel, and are there-
fore to be preferred. Arranged according to galactic
latitude these gauges give a star-density falling steadily
from 1375 stars per square degree at the galactic circle to
137 at the galactic pole. The limiting magnitude is
determined by indirect means to be 13 '9, and it will be
seen that the ratio 10 : 1 is practically equivalent to the
definitive result adopted by Kapteyn.

It appears, then, that the Herschel gauges are the main
source of the discrepancy; but the results were not accepted
without careful checks being applied. These were of two
kinds ; first, counts of stars were made on forty-five photo-
graphs, chiefly of the fields of variable stars, for which the
limiting magnitude on a visual scale could be calculated
from standard stars, photometrically determined. These,
when arranged according to galactic latitude, gave results

ix COUNTS OF STARS 193

in excellent accordance with the star-gauges ; and it was
considered that this checked the constancy of Herschel's
limiting magnitude. The remaining source of statistics

o o o

was provided by the published charts of the Carte du del
taken at Algiers, Paris, and Bordeaux. There is no means
as yet of determining the limiting magnitude of these
independently ; but, as it seems reasonable to assume
that any fluctuations will be accidental and have no
systematic relation to galactic latitude, they may be used
to obtain the ratio of galactic concentration. The ratio
found by Kapteyn between the belts of latitude 20 C
and 40" -90 was 5'5 : 1. For Zones I and VIII the
ratio would naturally be greater, and, moreover, the
numbers refer to a limit about one magnitude brighter
than Herschel's gauges. The ratio 10 : 1 at magnitude
14 is therefore supported by three independent sources of
evidence Sir J. Herschel's gauges, counts on variable
star fields, and counts on the French astrographic charts.

It has been suggested by H. H. Turner * that the
discordance is due to a real difference in the distribution
according as the magnitudes are reckoned visually or
photographically. The Herschel counts refer directly to
visual magnitudes, and the counts of variable star fields,
although made on photographs, are reduced in such a way
that the results refer to the visual scale. The counts on
the French chart-plates, however, relate solely to the
photographic scale ; and it is only by disregarding this
evidence that the suggestion reconciles the two results.
Perhaps we may consider the third source more doubtful
than the two former, for the plates are distributed only
through a narrow zone, and may be affected by abnormal
regions of the sky. The possibility of a real difference
in the galactic concentration for visual and photographic
results is an interesting one. It appears to mean that

* Who had also arrived at a small value of the galactic concentration for
photographic magnitudes (Monthly Notices, Vol. 72, p. 700).

O

i 9 4 STELLAR MOVEMENTS CHAP.

there are in the galactic regions vast numbers of faint
stars too red to be shown on the photographs. This may
be due either to a special abundance of late type stars in
the more distant parts of the stellar system, or more
probably to the presence of absorbing material a fog in
interstellar space, which scatters the light of short wave-
length.

The hypothesis requires further confirmation. A dis-
cussion by E. C. Pickering is directly opposed to it. He
also used the visual determinations of magnitude in the
fields of variable stars for his data ; but he applied them
more directly. His conclusion was that " the number of
stars for a given area in the Milky Way is about twice as
great as in the other regions and the ratio does not increase
for faint stars down to the twelfth magnitude 4 ."

One of the most interesting results of Chapman and
Melotte's investigation is that the total number of stars in
the sky for the fainter magnitudes is much smaller than
has often been supposed. Kapteyn's table gave 389
million down to 17 m> against 55 million according to the
present investigation. The excess of Kapteyn's numbers
is almost wholly due to his high value of the galactic
concentration ; the two investigations are practically in
agreement at the galactic poles.

By inspection of Table 32 it will be seen that the rate
of increase in the total number of stars has fallen off very
considerably for the last few magnitudes included. It
appears that the numbers are beginning to approach a
limit. An attempt to determine this limit involves a
somewhat risky extrapolation, yet the convergence has
already become sufficiently marked to render such extra-
polation not altogether unjustifiable. The empirical
formula \ogB m = a + 0m ym 2 cannot be pressed far
beyond the range for which it was determined, since it leads
to the impossible result that the number of stars down to a

ix COUNTS OF STARS 195

given magnitude would ultimately begin to diminish. By
a simple modification more suitable formulae can be

1 T>

obtained. Instead of using B m we consider b m = -

dm

that is to say, we use the number of stars of the magnitude
m instead of the number brighter than m. It is found
that an equally good approximation is obtained by setting

log ]0 6 )H = a + bm -cm 2 ......... (2)

and with this form the whole number of stars approaches
asymptotically to a definite limit as m is increased.
We have then

dm

(a+bm-cml),log u e

Cm ,

B, n = / 6 din

, ry*-> ^ .......

Vfl" J 'JO

where

A = /" logic*? m 10 a-f 62 /4c B = / _C_ Q = _&

V c V Iog 10 e 2c

From the formula (3) it is seen that A represents the
total number of stars of all magnitudes, and (7 represents
the median magnitude, i.e., the limit to which we must
go to include half the stars.

As c is not very easy to determine, two formulae may
be given between which the truth probably lies :

Iog 10 6 m = -0-18 +0720 m -O'OlGOm 2 . . (4)
Iog 10 &m = +0-01 +0-680 m -0'0140m 2 . . (5)

We have here altered the unit of area, so that b m refers
to the number of stars in the whole sky instead of to a
square degree.

These formulae lead to

(4) (5)

Whole number of stars of all magnitudes 770 millions 1800 millions
Median magnitude .......... 22 m '5 24 m> 3

Chapman and Melotte conclude : " Unless the general
form of our expression for B m ceases to apply for values

o 2

196 STELLAR MOVEMENTS CHAP.

of m greater than 17 (up to which the accordance is good)
it is possible to say with some probability that half the
total number of stars are brighter than the 23rd or 24th
magnitude, and that the total number of stars is not less
than one thousand millions and cannot greatly exceed
twice this amount/'

The mean star-density for given galactic latitude and
limiting magnitude has been given in Table 31. A
question arises as to how far these are sufficient to determine
the star-density at any particular spot, and what variations
from the mean are likely to arise. The possible variations
may be classed as follows.

(1) A systematic difference between the north and south
galactic hemispheres.

(2) A systematic dependence on galactic longitude in
some of the zones.

(3) General irregularity.

There is not a very conspicuous difference between the
two galactic hemispheres. So far as the tenth magnitude
the southern hemisphere is found to be the richer by 10 or
15 per cent. ; for fainter stars no difference is found. We
have to depend for this conclusion on researches made
before the introduction of modern standard magnitudes,
but the evidence seems to be satisfactory. To account for
the small difference in richness it has been usual to suppose
that the Sun is a little north of the central plane of the
stellar system ; this agrees with the appearance of the
Milky Way, which deviates slightly from a great-circle,
and has a mean S. Galactic Latitude of 17,

It is the opinion of most investigators that, except in
the Milky Way itself, the differences depending on
galactic longitude are inconsiderable ; galactic latitude
is the one important factor and overshadows all other
variations. This view seems to be l>;is<>l on ;i ^cimral
impression rather than on any quantitative results. A

ix COUNTS OF STARS 197

detailed investigation is greatly to be desired, for the
conclusion must still be considered open to question. The
following calculation appears to furnish an upper limit
to the possible variations of the limiting magnitude.

For the majority of the Franklin- Adams chart-plates,
unstandardised counts have been given by Chapman and
Melotte. Taking, for example, the Johannesburg plates,
the centres of which lie between galactic latitudes 20 29
and 30 39 , we find the following results, which refer to
a limiting magnitude about 17 ni '5 :

Zone. 20 2 29 30 39

Number of Plates 20 16

Smallest number of stars per plate . . 292,000 . 306,000

Largest ,, . . .737,000 577,000

Average deviation of log density from

the mean 0'078 0'059

Corresponding ratio 1*20 : 1 1'15 : 1

The average deviation (20 and 15 per cent, respectively)
includes not only the actual fluctuations of star-density,
but variations due to the quality of the plates and the
personality of the counters. Its smallness bears witness
to the uniformity of the Johannesburg sky as well as
to the regularity of stellar distribution.

There is no doubt that within the limits of the Milky
Way very considerable variations of star-density occur.
The most remarkable region is in the constellation Sagit-
tarius, where certain of the star-clouds are extraordinarily
rich. This part of the Milky Way is unfavourably placed
for observation in the latitude of the British Isles ; but
from more southerly stations it appears the most striking
feature of the heavens. It was found that on the
Franklin-Adams plates the images of the faintest stars
in this region were so close as to merge into a continuous
background and it was impossible to count them. We
should expect that the presence of the Milky Way
clusters would add great numbers of stars beyond the
normal increase towards the galactic plane, and it is

198 STELLAR MOVEMENTS CHAP.

rather surprising that there is not a more marked dis-
continuity between the numbers for Zone I and Zone II
in Table 31. Probably the dark spaces and tracts of
absorbing matter, which are a feature of the Milky Way,
neutralise the effect of the rich regions, and bring about a
general balance.

The star-ratio falls a great deal below its theoretical
value 3 '9 8 for an infinite universe even as early as the
sixth magnitude. The magnitude-counts are not, however,
sufficient by themselves to determine the rate at which
the stars thin out at great distances. For this we need
additional statistics of a different kind, as will be shown
in the next chapter. Meanwhile, although the star-counts
do not determine a definite stellar distribution, we can
examine whether any simple form of the law of stellar
density in space is consistent with them. The simple
result that the galactic concentration is independent of
the magnitude, even if it were rigorously true, would not
admit of any correspondingly simple interpretation in
terms of the true distribution in space. It is therefore of
no help in our discussion.

Consider a stellar system in which the surfaces of equal
density are similar and similarly situated with respect
to the Sun as centre, the density falling off from the inner
parts to the outer. We naturally think of spheroids of
the same oblateness.

Let the ratio of the radii towards the galactic pole and
in the galactic plane be 1 : v.

Corresponding to an element of volume S at a distance
r towards the pole, there will be an element of volume v 3 S
at a distance vr in the galactic plane, containing stars
distributed with the same density. The number of these
stars, being simply proportional to the volume, will be ^
times as many in the second case, and this will apply to
all grades of intrinsic brightness separately. But their

ix COUNTS OF STARS 199

apparent brightness will be diminished in the ratio v~- 9 or,
expressed in magnitudes, they will be 5 log v magnitudes
fainter. This holds for all the elements of volume S in a
cone from the Sun to the galactic pole, and the correspond-
ing elements v s S in a cone from the Sun to the galactic
equator.

Hence if the number of stars brighter than a given
magnitude is given by

B m = fy(m) for the galactic pole

it will be given by

B m = v 3 \lr(m - 5 Iog 10 v) for the galactic plane.

Now we have found (Table 33) that for the galactic
pole

\oB m = 0-901 +0-380 (m- 11) -0'013 (m-11) 2 .

Hence for the galactic plane

logB m = 31ogx +0-901 -0-380 x51ogi/ -0'013 (5 log*)-

+ (0-380 + 0-013 x 10 log v) (m - 11) -0'013 (i - II) 2 .

If log v = 0'54, this reduces to

log# w = 1-400 +0-450 (in- 11) -0'0130 (m- II) 2 ,

which is fairly close to the true value for Zone I, viz.

log.B m = 1-404 +0409(m-ll) -0-0139 (MI- 11)-.

The excess of the calculated coefficient 0*450 over its
observed value can be interpreted to mean that the
diminution of density in the galactic plane is rather more
rapid than would be the case if the surfaces of equal
density were similar. The oblateness of the distribution
becomes less pronounced at great distances. The differ-
ence is not due to accidental error, for it can be shown
that any other pair of zones would have given a greater
excess in the same sense. The oblateness v is therefore
not a constant quantity ; and the above value corresponds
to a certain average distance which may be roughly
expressed as the distance of the stars of the eleventh
magnitude.

200 STELLAR MOVEMENTS CH. ix

For logi> = 0'54, we have v = 3'5. This, then, is the
average oblateness, or ratio of the axes of the surfaces of
equal density, v must not be confused with the galactic
concentration of the stars ; it is a pure accident that their
values are nearly the same.

REFERENCES. CHAPTER IX.

1. Chapman and Melotte, Memoirs, R.A.S., Vol. 60, Pt. 4.

2. Kapteyn, Groningen Publications, No. 18.

3. J. Herschel, Results of Astronomical Observations at the Cape of Good
Hope, Cp. iv.

4. Pickering, Harvard Annals, Vol. 48, p. 185.

CHAPTER X

GENERAL STATISTICAL INVESTIGATIONS

IN the application of mathematics to the study of
natural phenomena, it is necessary to treat, not the actual
objects of nature, but idealised systems with a few well-
defined properties. It is a matter for the judgment of the
investigator, which of the natural properties shall be
retained in his ideal problem, and which shall be cast
aside as unimportant details ; he is seldom able to give a
strict proof that the things he neglects are unessential, but
by a kind of instinct or by gradual experience he decides
(sometimes erroneously it may be) how far his representa-
tion is sufficient.

In the idealised stellar system, which will now be
considered, there are three chief properties or laws. The
determination of these must be regarded as the principal
aim of investigations into the structure of the sidereal
universe, for if they were thoroughly known we might
claim a very fair knowledge of the distribution and move-
ments of the stars. The laws are :

(1) The density law. The number of stars per unit
volume of space in different parts of the system.

(2) The luminosity law. The proportion of stars
between different limits of absolute brightness.

(3) The velocity law. The proportion of stars having
linear velocities between different limits both of
amount and direction.

201

202 STELLAR MOVEMENTS CHAP.

As regards the first of these, the density may be assumed
to depend on the distance from the Sun and on the galactic
latitude. The decrease of density at great distances from
the Sun represents the fact that the stellar system is
limited in extent, and as it is notorious that the limits are
very much nearer towards the galactic poles than in the
galactic plane, a representation which did not include a
variation with galactic latitude would be very imperfect.

The luminosity law and velocity law may be assumed
tentatively to be the same in all parts of space. Argu-
ments may be urged against both these assumptions ; but
they seem to be inevitable in the present state of knowledge,
and it is probable that the results obtained on this basis
will be valid as a first approximation. It may further be
remarked that in dealing with proper motions we are
necessarily confined to a rather small volume of the
stellar system, and the assumption of a constant velocity
law in such investigations seems to be justified.

The constancy of the velocity law is in most investiga-
tions assumed in a different form, which must be carefully
distinguished from the assumption just stated, and is, in
fact, much less innocuous. It is assumed that for the
stars of a catalogue the velocity law is the same at all
distances. Now among the stars of a catalogue with a
lower limit of magnitude there is a strong correlation
between luminosity and distance. Thus an additional
assumption is virtually made, that the stars of different
intrinsic luminosity have the same velocity law ; or, since
spectral type and brightness are closely associated, that
the stars of different spectra have the same velocity
law. This is well-known to be untrue. It seems likely
that results obtained on this assumption (including some
investigations in this chapter) may be misleading in some
particulars ; though here again we may often arrive
at fairly correct conclusions notwithstanding imperfect
methods. But it is desirable, where possible, that the

x STATISTICAL INVESTIGATIONS 203

different spectral types should be investigated separately :
for in the case of stars of homogeneous type we know of no
evidence to invalidate the hypothesis of a constant velocity
law.

A further property of the stellar system, which has some
claim to be retained in the -idealised representation, is the
absorption of light in inter-stellar space. There is some
evidence, insufficient it may be, that this is small enough
to be neglected in the present discussions. As it does not
seem practicable to deduce useful results when it is retained
as an unknown, we shall take the risk of neglecting it.

The problem of determining one or more of the three
laws that have been enumerated may be attacked in various
ways ; and the diversity of the investigations is rather
bewildering. The difficulty of giving a connected account
of the present state of the problem is increased by the
fact that some of the work has been based on data that are
now obsolescent ; and it is difficult to know how far the
introduction of more recent figures would produce import-
ant modifications.

The general statistical researches described in this
chapter depend on one or more of the following classes
of data :

(a) Counts of stars between given limits of magnitude.

(b) The mean parallactic motion of stars of given mag-
nitudes.

(c) Parallaxes measured directly.

(d) The observed distribution (or spread) of the proper
motions of stars brighter than a limiting magnitude.

The radial velocities are only appealed to for the adopted
speed of the solar motion, usually taken as 19*5 km.
per sec.

It is convenient to distinguish the use of the proper
motions for determining the mean parallactic motion from
other applications of the proper motion data. The paral-
lactic motion, or as it is sometimes called, the secular

204 STELLAR MOVEMENTS CHAP.

parallax, fixes the mean distance of a class of stars
without introducing any consideration of the distribution
of their individual velocities.

The investigations may be grouped into three classes :

I. Those which depend on (a) and (b) only.

II. Those which depend on (a), (6), (c), and (d).

III. Those which depend on (d) only.

It will be shown later that (a) and (b) are theoretically
sufficient to determine the density and luminosity laws,
so that the inclusion of (c) introduces some, redund-
ancy of equations. Investigations depending on (d) stand
rather apart from the others, since they involve the velocity
law ; but as they also throw some light on the other two
laws, it is useful to consider them in the same connection.

Before proceeding to consider the three classes of
investigation, it is necessary to give some attention to the
expression of the results for the mean parallactic motions
and the measured parallaxes in a suitable form. The
formulae given by J. C. Kapteyn in 1901 are very widely
used ; and, although it would naturally be an improve-
ment to substitute more recent data, no general revision
of his work has yet been made. Kapteyn derived two
formulae 'for the mean parallaxes of stars ; one
expressing the mean for all stars of a given magnitude,
the other for stars of given magnitude and proper motion.
For the first formula it is not possible to make use of the
directly measured parallaxes. Jt is not so much that the
data are too scanty, as that the stars are usually selected
for investigation on account of large proper motion, and
are accordingly much nearer to the Sun than the bulk of
the stars of the same magnitude. The mean parallactic
motion, or secular parallax, provides the necessary means
of determining the dependence of parallax on magnitude
alone. The only practical difficulty arises from the
occasional excessive motions, which exercise a dispropor-

STATISTICAL INVESTIGATIONS

205

tionately great influence on the result. Kapteyn's results,
which depend on the Auwers-Bradley proper motions, are
contained in the formula,

Mean parallax for magnitude m = 0""0158 x

(A)

If Types I and II are taken separately, the formula?
for the mean parallaxes are :

Type I 7? m - 0"'0097 x (078)-5-5.

,, II 7f w = 0"-0227 x (078)-5'5.

In Table 34 the mean parallaxes for the different
magnitudes are given, after correcting the foregoing
formula (A) to reduce to the modern value of the solar
motion, 19*5 km. per sec., instead of 16*7 used by
Kapteyn.

TABLE 34.

I\<i/>teyns Mean Parallaxes.
(Reduced to the value 19*5 km. per sec. for the solar motion.)

Magnitude.

Mean Parallax.

Magnitude.

Mean Parallax.

1-0

0-0414

6-0

0-0120

2-0

0-0323

7-0

0-0093

3-0

0-0252

8-0

0-0073

4-0

0-0196

9-0

0-0057

5-0

0-0153

For the dependence of parallax on proper motion,
Kapteyn had recourse to the measured parallaxes. For
this purpose their use is quite legitimate, though we may
be inclined to doubt whether the data (at that time much
less satisfactory than now) were sufficiently trustworthy.
For a constant magnitude, the dependence on the proper
motion p was found to follow the empirical formula

where _p=l,r405. The actual formula giving the

206 STELLAR MOVEMENTS CHAP.

mean parallax of a star of magnitude in and proper
motion rf' per annum is

tm,n. = (0-906)-5-5 x (0-0387/i)' n2 ' (B)

It is interesting to know how nearly this mean formula
is likely to give the correct parallax of a particular star.
Assuming that log(7r/7r) is distributed according to the law
of errors, the probable deviation of this logarithm has
been found to be 0'19. Thus it is an even chance that the
parallax of any star will lie between 0'65 and 1*55 times the
most probable value for the given motion and magnitude.*
This result was found by comparing measured parallaxes
with the formula, and determining the average residual.

The formulae for 7r m and ir m ^ may be more conveniently
expressed in the form t

Iog 10 irm = - M08 -0-125m, (C)

log 10 7r, M = - 0-766 -0'0434m +0-712 log /x ' (D)

In order to bring together all the data due to Kapteyn,
we may add here his results for the counts of stars of
successive magnitudes. It has been shown by K. Schwarz-
schild that Kapteyn's numbers can be summarised by the
formula :

Iog 10 6, n = 0-596 + 0-5612 m- 0-0055 m 2 (E)

where b m is the number of stars (in the whole sky) between
magnitude m and m +dm.

The formulae (C), (D), and (E) correspond respectively
to the data (6), (c), and (a) previously mentioned.

The main criticism, of subsequent investigators has
been directed against the large value of the coefficient of
m in formula (C). There is some reason to believe that
the decrease of mean parallax with increasing faintness is
less rapid than is shown in Table 34. According to C. V.

* The most probable value is not the mean value. With the above prob-
able deviation the most probable value would be 0'81 x 7r m , M .

f Formula (A) and (C) do not quite correspond as the former contains a
slight correction made by Kapteyn (Preface, Groninyen Publications, No. 8).

x STATISTICAL INVESTIGATIONS 207

L. Oharlier 1 the coefficient is less than half Kapteyn's
value. Charlier's determination rests on the Boss proper
motions, which are of great accuracy ; but the range of
magnitude is too limited for a satisfactory solution.
Without attaching much weight to the precise value, he
considers that Boss's proper motions cannot be reconciled
with the larger figure. This means that there is less
difference between the parallaxes of different magnitudes
than Kapteyn supposed. The writer, 2 also working on
Boss's Catalogue, had found that for stars of a given
magnitude the spread in distance must be less than that
deduced from Kapteyn's formulae, a fact which is evidently
related to Charlier's objection.

G. C. Comstock 3 has likewise maintained that the paral-
laxes of faint stars are larger than those given by the
formula. From an investigation of the proper motions of
479 stars from 7 m to 13 m he concluded that a relation,
first given by A. Auwers for a shorter range, holds satis-
factorily from the third to the thirteenth magnitudes,
viz., that the mean proper motion is inversely proportional
to the magnitude. As the mean proper motion may be
taken to be proportional to the parallax, Comstock's result
leads to the formula

TT = c/m (in >3).

The formula (E) may be compared with Chapman and
Melotte's determination (formulae (4) and (5), p. 195).
The real differences are very large, though perhaps not
quite so great as would appear from a cursory comparison
of the coefficients.

I. INVESTIGATIONS WHICH DEPEND ON COUNTS OF STARS
AND ON MEAN PARALLACTIC MOTIONS.

From the mean parallaxes of stars of different magni-
tudes combined with the counts of the number of stars
down to limiting magnitudes, the density law and the

208 STELLAR MOVEMENTS CHAP.

luminosity law can be determined. A most elegant gene-
ral solution of this problem has been given by Schwarz-
schild ; and, although it may usually be less laborious in
practice to work out special cases according to the func-
tions which represent the observed data, his method is so
generally applicable to the fundamental problems of
stellar statistics that we shall begin by considering it at
length.

Let D(r) be the number of stars per unit volume at a
distance r from the Sun.

Let $(i) di be the proportion of which the absolute
luminosity lies between i and i + di.

Setting h for the apparent brightness of a star, we have

Let B(h) dh be the total number of stars of apparent
brightness between h and h + dh.

And let ir(h) be the mean parallax of stars of apparent
brightness h.

o

Then the whole number of stars at distances between
r and r + dr is

. D(r),

and of these the proportion <(Ar 2 ) r 2 dh will have an
apparent brightness between h and h + dh.
Hence

H(h)dh =
or

B(h) =

And for the sum of the reciprocals of the distances

We now transform the two integrals (2) and (:*) as
follows :

x STATISTICAL INVESTIGATIONS 209

Let

r = e ~ P, h = e- 2 M,

so that

i = e ~ 20* + P).

Further let

60*),

Here, since .!?(/&) and TT(^) are supposed to be given by
the data of observation, b(fi) and c(^) are given likewise ;
they are the same observed quantities expressed as func-
tions of a changed independent variable.

The two integral equations (2) and (3) become

&GO= f(p)9(t*+p)dp ................ (4)

J -x

c( M )= / f(p)g( f j L +p)e P dp ............... (5)

.' -30

Let us form the Fourier integrals

*() = -i- r

5!rJ -oo

K) = -I/'" / M )e-.^ ( )-- L|" flOOe-*da

27T.' -ao 27JV -oo

where t= ^/ 1.

Then we have the well-known reciprocal relation

J -00 J - 9

. x J -

Now

i /" x / x

= o-/ /

7TJ -x J -

= ^/" f(p

-STTj - x

Hence by (6)

. . (8)
P

eiqp

-oo

210 STELLAR MOVEMENTS CHAP.

Similarly,

-j /

<(</) = o- I

^TJV -o

= 9(9) r

J -x

Thus

cfa) = 2irf(i- 9)9(9) ................... (9)

From (8) and (9) we have

fi=i> = <i> . (10)

!(-) K)

As the functions c and fc can be calculated directly (by
Fourier analysis or otherwise) from c and b, the right-
hand side is known.

Setting

p = iq

and

equation (10) becomes.

F(^ + l)-^) = log^|) .............. (11)

a difference equation of which the solution is

the integration being along the imaginary axis of p'.

Thus F can be found, and from it f. Then f is deter-
mined by means of (7).

Further, when f has been determined, g is given by
(8) and then g can be determined.

The density and luminosity laws are accordingly found.

If particular forms are assumed for the expressions
which give the counts of stars of each magnitude and their
mean parallaxes, the analysis may be made much simpler.
In the following investigation special forms of the functions
are adopted, which appear adequate to represent the present

x STATISTICAL INVESTIGATIONS 211

state of our knowledge, and lend themselves conveniently
to mathematical treatment :
Let

r be the distance measured in parsecs,

/ the absolute luminosity measured in terms of a star of zero magnitude

at a distance of one parsec ,
h the apparent brightness in terms of a star pf zero-magnitude.

And set

p = -5'01og 10 r,
M= -2-51og 10 i,
m = 2*5 Iog 10 #,

where M and m are accordingly the absolute and apparent
magnitudes.

We have i = hr 2 , and M = m + p.

We adopt the forms :

Density law: D(r) = 10o - "IP - 2P 2 .......... (13)

Luminosity law : </>(i) = 10 ft o - &i-V - &2-V2 ......... (14)

Number of stars between magnitudes m and m + dm :
6(m)(Jm = 10- !-** dm ........ (15)

Mean parallax of stars of magnitude m

7r(r/i) = 10 p o--Pim-P 2 m2 ......... (16)

Then, expressing that the stars of magnitude m to
m + dm are made up from successive spherical shells at
distance r, containing 4?ir 2 dr D(r) stars, of which the
proportion < (/*/'") /' 2 dh are of the appropriate intrinsic
brightness, we have

(17)

Now

dm = - 2'5 Iog 10 e dh h,
dp = -5'01og 10 e drjr,

r = 10- 0"2 P

and

(r) = 10&o - *i(m + P) - Wm + p)2 + oo - o 1P - aapS^

p 2

212 STELLAR MOVEMENTS

Hence we have b(m) =

dp . 10-p-' 4Bl + 0-lP-2P 2 +

CHAP.

. . (18)

12-5 (log e) 2 J -a,

Now the integral is of a well-known form,

r dp !CMo-

J -oo

The reduction evidently leads to an expression for b(m)
of the form set down in (15), and we find

(19)

The sum of the parallaxes of stars between magnitudes
ra and m + dm, which is equal to ir(m) b(m)dm, is found
by writing r z for r* in (17), or ir(m) b(m) is given by writing
^-0*2 for a^ in (18) and (19). Carrying through this
change in (19), we find

(20)

+P _

Hence subtracting (19) from (20)

(21)

The fact, that quadratic formulae for the logarithms of
the density and luminosity functions lead to a linear
formula (P 2 = 0) for the logarithm of the mean parallax
is of interest, because the linear formula for the latter
is the one given by Kapteyn, and is in general use.

From the formulae (19) and (21) it is easy to deduce the

x STATISTICAL INVESTIGATIONS 213

coefficients of the density and luminosity functions in
terms of the observed coefficients * , K l9 * 2 , P , and P lt

Thus a 2 = I, , If - = :. , and so on. a and b are not
O/i 1

given independently, but as a sum a + 6 . But, if desired,
6 can be found from the condition implied in the definition
of

Of practical attempts to determine the density and
luminosity functions from the mean parallaxes and
counts of stars of different magnitudes, an investiga-
tion by H. Seeliger 4 (1912) may be taken as .an example.
Dividing the sky into five zones according to galactic
latitude, he obtained for B, n (the number of stars per
square degree down to magnitude m) the following
expressions :

Zone. Galactic Latitude. Formula.

A 90" to 70 log lo B Ifl = -4-610 +0*6640 m - '01334 m 2

B 70 c to50 -4-423 + 0'6099 m - "00957 m 2

C 50 : to30 ,, -4-565 +0'6457 m - '01025 m 2

D 30= to 10 = -4-623 +0'6753 m -0-01027 m 2

E + 10' to- 10 ' ,, -4-270 +0-6041 m -0 '00512 m 2

These were derived from a discussion of the magnitudes
of the Bonn Durchmusterung and from Sir John Herschel's
gauges. Owing to the use of the latter source, the
galactic concentration of the faint stars is very strong,
just as in Kapteyn's results ; the doubt cast on these
numbers by modern researches has been fully discussed in
the preceding chapter. The course of Seeliger's coefficients
from zone to zone appears quite irregular, but the form of
the expressions tends to conceal a steady progression in
the actual numbers.

From these expressions for B m , b m = ^ was deduced

dm

without difficulty and expressed in the same quadratic
form (equation 15). For the mean parallaxes (equation

2i 4 STELLAR MOVEMENTS CHAP.

16) Kapteyn's numbers were adopted. But a difficulty
arises because the meau parallaxes have been given only
for the sky as a whole, and not for the separate zones of
galactic latitude ; and it is well known that the mean
parallax varies greatly from one latitude to another. The
difficulty, however, can be surmounted. We have agreed
to regard the density law as depending on the galactic
latitude, and the luminosity law as constant ; that is to
say, the coefficients , a lt a 2 will be different for each
zone ; b , b lt b. 2 will be the same for all. If we eliminate
! and a 2 from (19) and (21), and disregard the first
equation, which is only used for determining a + 6 , we
obtain two equations between b ly b. 2 and P , P 1 from each
zone. Combining these according to the number of stars
in each zone, two equations are obtained in which it is
permissible to substitute the mean P l and P given by
Kapteyn's parallaxes. These determine b l and b. 2 . The
remaining constants a + b , a lt a 2 are then determined for
each zone separately by means of equations (19). By a
procedure of this kind, though differing in detail, Seeliger
arrived at a solution. That the result is very different
from what would be obtained by using the same parallax
formula for all zones may be seen by the following
numbers, which were deduced as the mean parallaxes of
stars of magnitude 9 m< 0.

Zone. A. B. C. D. E.

Mean Parallax (9 m '0) (T0065 0"0061 CT0053 0"'0044 0"'0039

The density and luminosity functions are given by (13)
and (14). Seeliger's result for the luminosity function is

#t) = const, x*- 2 ' 12910 ^- ' 1007 ' 10 *'* (22)

As examples of the law of density the following (given
iv Seeliger) may suffice:

Zone. A. B. 0. D. E.

Density at distance 5000 parsecs Q . mi ^^ Q . 0355

Density at the bun
Density at 1600 parsecs . Q21 ^^ . 1Q7

Density at 10 parsecs

x STATISTICAL INVESTIGATIONS 215

These results show a much more rapid falling off in
density near the poles compared with the galactic plane.
The results for Zone E should perhaps be set aside, since
presumably they are disturbed by the passage of the Milky
Way through that region ; but the other four zones repre-
sent the general distribution in the stellar system. We
do not, however, place much reliance on the numerical
results, since the determination rests on the Herschel
gauges and on Kapteyn's mean parallaxes, both of which
are open to some doubt.

II. INVESTIGATIONS WHICH DEPEND ON COUNTS OF
STARS, THE MEAN PARALLACTIC MOTIONS, MEASURED
PARALLAXES AND DISTRIBUTION OF PROPER MOTIONS.

Suppose that a table of double-entry is formed giving
the number of stars between given limits of magnitude
and given limits of proper motion. For the stars in any
compartment of the table corresponding to magnitude ra
and proper motion p, the formulae (B) or (D) give the
mean parallax. Further, as already stated, the individual
parallaxes deviate from the mean according to the law,

Frequency of log (TT/TT) is an error function with probable error 0'19.

Hence the proportion of these stars between any
given limits of parallax can be found. Thus, taking
the stars in any one compartment, we can redistribute
them into a new table with arguments parallax and
magnitude. Treating each compartment of the old table
separately, we transfer all the stars into the new table,
and obtain the number of stars between given limits of
magnitude and of parallax.

Table 35 gives the results of a solution made in this
way by J. C. Kapteyn. 5 It shows how the number of
stars of each magnitude are distributed as regards dis-
tance.

216

STELLAR MOVEMENTS

CHAP.

TABLE 35.
in Distance of Stars of each Magnitude (Kapteyn).

Limits of Mean
Distance. Parallax.

Number of Stars in the Sky.

m-M.

T> //

m.

m.

m.

m

m. 1 m.

Parsecs.

2T,-3r.

3-6-4-6

4-6 -5-G

5-6-6-6

6-6-7-6

7-6-8-6

m.

>1000

0-6

5-0

25

127

703

4840

631-1000 0-00118

2-0

8'0

42

197

871

4590

14-5

398 -631 0-00187

2'9

19-6

92

369

1466

6050

13-5

251398 ; 0-00296

9-4

29-6

151

603

2210

7310

12-5

158251 0-00469

14-7 51-0

223

815 2770

8320

11-5

100158 0-00743

19-6

64-6

256

885 2760

5830

10-5

63100 ! 0-0118

22-8

72-8

240

767

2080

4140

9-5

4063 0-0187

21-3

71-1

190

537

1240

2150

8-5

2540 0296

17-2

57-1

130

311

579

890

7-5

1625 0-0469

11-8

39-1

71

145

235

316

6-5

1016 0-0743

6-5

225

34

56

84

99

5-5

6-3-10 0-118

3-2

11-2

14

18 29

30

4-5

06-3

2-0

7-0

8

11 14

14

For stars of known magnitude and distance the abso-
lute magnitude M can be calculated. The number to be
subtracted from the apparent magnitude m to give the
absolute magnitude will be found in the last column of
the table. The limits of distance were so chosen that
the step from one line to the next corresponds to a change
of one magnitude.

Each line of Table 35 gives a determination of the
luminosity law, for it exhibits the number of stars in a
certain volume of space which have absolute magnitudes
between given limits. By taking suitable means between
the results from each line of the table, Kapteyn arrived
at an expression for the luminosity law which may be
written

const x e - 1 ^ log* < - o-07-j (log, i)3

(23)

This may be compared with Seeliger's result (22).

Again, if we start from any number on the left side of
Table 35, and move diagonally upwards to the right, the
successive numbers will all refer to stars of the same

x STATISTICAL INVESTIGATIONS 217

absolute magnitude. For example, starting from 17*2, we
have the numbers

17-2 7M 240 885 2770 7310

referring to an absolute magnitude 4*4 (strictly 4*9
to -3'9).

Now these are the numbers of stars in a series of
spherical shells, the volumes of which form a geometrical
progression

1 4-0 15-8 63-1 251 1000

Whence by division the relative densities are

17-2 17-8 15-2 14-0 11/0 7'3

corresponding to distances of

25-40 4063 63100 100158 158251 251398 parsecs.

According to our hypotheses, the change of star density
with distance will be shown equally whatever absolute
magnitude is selected ; we shall thus obtain from Table 35
a number of determinations of the density law. The
following numbers will illustrate the character of the
density law deduced by Kapteyn.

Distance. Star-density.

1-00

50 parsecs 0'99

135 ,, 0-86

213 0-67

540 0-30

850 0-15

It has been mentioned that in this second class of
investigation, more data are employed than are necessary
to give a solution. That is why we obtain from Table 35
a number of separate determinations* of the luminosity
and density laws instead of a single solution. Kapteyn
has used the agreement of the separate determinations to
defend the assumption that the absorption of light in
space is not large. Schwarzschild has discussed the
mutual consistency of the data analytically and has
shown that the theoretical relations are well-satisfied.

2i8 STELLAR MOVEMENTS CHAP.

Thus he finds that the theoretical value of the probable
error of log ir/ir is 0*22, compared with 0'19 adopted
by Kapteyn from the observations.

IH. INVESTIGATIONS WHICH DEPEND ONLY ON THE
DISTRIBUTION OF THE PROPER MOTIONS OF STARS.

Another class of statistical investigations depends
entirely on the proper motions. In this it is convenient
to introduce a new definition of the density law, viz., the
number of stars brighter than a limiting apparent
magnitude per unit volume of space at different distances
from us. This involves a combination of the old density
and luminosity laws, and in itself gives us no definite
information as to the structure of the system ; but it is
clearly a matter of great practical interest to know how
the stars of our catalogues are distributed in regard to
distance. The determination of the velocity law is also an
object of these investigations.

Let

g(u) be the number of stars having linear motions between u and
u

u + duj
h(a) the number having proper motions between a and a + da ;

/(r) the proportion of stars (of the catalogue) at a distance between
r and r + dr from the Sun.

Then, expressing that h(a) ' l is made up of stars at all

a

possible distances r, with linear motions u between ra and
r(a + da).

*(.)*? = r&^V^rda.
a J / ra

Hence

Or writing

u = ey

x STATISTICAL INVESTIGATIONS 219

Equation (24) becomes

IV =/" f(X) 8 (X+/i)dX ............. (25)

J -oo

This is of the same form as (4), and the solution is
accordingly

H(,j) = 2irF(-q)G(q) ............. (26)

where F, G, H are the Fourier integrals corresponding to

............. (27)

&irJ oo

and

f(X) = T F(q)e^dq ............... (28)

To obtain a starting point it is assumed that in the
direction at right angles to the vertex of star-streaming
the linear motions are distributed according to the error-

_/ </ -( ? 4__ y\-
law e ' du, where F is the component of the solar

motion in that direction. The evidence for this has been
discussed in Chapter VII. There is the advantage that
both the two-drift and ellipsoidal theories agree on this
point, so that it is a fair assumption in any comparison of
the t\vo theories. Confining attention to the stars moving
in this direction, g can be found, and hence G is deter-
mined by periodogram analysis. For the same direction,
H is determined by an analysis of the observed motions,
and then F is given by (26). Having once found F
(which determines the density law f), we may use its
value in (26) to determine G for other directions, and
g is determined from G by another periodogram analysis
(equation (28)).

Thus from the one assumption that the proper motions
at right angles to the direction of star- streaming are
distributed according to Maxwell's law, it is possible to
determine the complete velocity-law. The method is a
little difficult to apply, not only on account of the long

22O

STELLAR MOVEMENTS

CHAP.

numerical computations, but also because the formulae,
which express statistical truths, cannot be exactly satisfied
by observations depending on a limited sample of stars.
The formal solution, which is unpleasantly conscientious,
knows only one way out of such a difficulty ; it diverges.
We therefore have to smooth the observed data before-
hand so as to make sure that the task is not an impossible
one ; and even then, if a trifling irregularity is left in, it

Solar
Antapex

Scale

20 30 40 50 Km. per sec.

FIG. 21. Curves of Equal Frequency of Velocity.

may cause the solution to make the most astounding
oscillations in the attempt to follow it exactly.

This method has been applied to 1129 stars, the proper
motions being taken from Boss's Preliminary General
Catalogue. The stars are in two opposite areas of the
sky with centres at the two Equinoctial points. As these
centres are nearly 90 from the vertex and from the solar
apex, the star-streaming and solar motion are shown in
this region without foreshortening.

The resulting distribution of the linear velocities is
shown in Fig. 21, which shows the curves of equal
frequency. Setting the number of stars with component

x STATISTICAL INVESTIGATIONS 221

velocities between x and x + dx, y and y + dy proportional
to ^(x,y) dxdy, the curves ^(x,y) = 1, 2, 3, etc., have been
traced. In the neighbourhood of the origin the present
method of calculating ty becomes indeterminate ; accordingly
within a certain distance of the origin the equi-frequental
lines have not been calculated, but the diagram has been
completed by the dotted lines, for it is sufficiently obvious
how the joining up must take place. "With the origin of
co-ordinates shown in the figure the velocities are referred
to the Sun ; they may be referred to any other standard by
merely changing the origin.

It will be seen that the diagram lends some support to
the hypothesis of two distinct drifts as opposed to the
unitary ellipsoidal hypothesis. It is not clear how far
this indication of a division is to be trusted. I do not
think that any modification of the analytical method or of
the original assumptions would make any difference in
the result ; but the data may be scarcely sufficient.

As F(q) is determined in the course of the analysis we
may also derive the distribution of the stars in regard to
distance. The results are as follows :

TABLE 36.

Distribution of the Stars of Boss's Catalogue (for a mean Galactic
Latitude 63).

Distance.

Parallax. Percentage of Stars.

Parsecs.

10-0 12-5

0-lQ 0-08

1-0

12-5 167

0-08 0-06

1-5

167 25-0

0-06 0-04

2-9

25-0 50-0

0-04 -0-02

9-5

50-0 667

0-02 0-015

8-6

667 100

0-015 O'OIO

17-5

100 125

0-0100-008

117

125 - 167

0-0080-006

14-9

167 - 250

0-0060-004

16-6

250 -- 500

0-004-0-002

10-0

500 1000

0-0020-001

2-4

222 STELLAR MOVEMENTS CHAP,

No less than 70 per cent, of these stars are between
50 and 250 parsecs distant from us. It is an even chance
that a 'star's parallax is contained within the limits 0"*012
and 0"'004 ; in other words, if we know no more of a star
than that it is brighter than 6 m '5, ive may set down iU
parallax as 0"'008 0"'004.*

In the foregoing investigation special attention wag
devoted to the calculation of the velocity-distributioc
with as little assumption as possible, and the distance -distri-
bution was found rather as a bye-product. We shall now
consider researches in which the leading object is to find
the distance-distribution.

Consider as usual the stars in an area of the sky
sufficiently small to be treated as plane.

Let the proper motions be resolved into components
along the direction towards the vertex and at right angles tc
it (in the plane of the sky). Denote the latter components
by T). t

Let

- V= Component of the solar motion (in linear measure) in th<

T} direction.
h(r)) drj = Number of stars whose component rj of proper motion liei

between 17 and rj + drj.
g(v) dv = Proportion of stars whose corresponding component o

linear motion lies between v and v + dv.
/(r) dr = Number of stars (in the region) at a distance r to r + dr ; sc

that the density of the stars brighter than the limiting

magnitude at a distance r from the Sun is proportional t<

r- 2 /(r).

Let further

H(rj) = \h(rf)dri = Number of stars whose component prope:
J o
motion is between and 17.

f\r) = / f(r) dr'= Number of stars whose distance is less than r.

* This remark is strictly true according to the ordinary definition o
probable error, but in a skew distribution of frequency such as this th<
probable error has not the pmjuTtirs ^ve usually associate with it.

t The notation of the piwinu.*, investigation is not conveniently applicabl
to this. We make an entirely fresh start.

x STATISTICAL INVESTIGATIONS 223

The units of r, v, and rj are made to correspond, so
that

i? = r;.

It is not difficult to see that the number of proper
motions greater than ij is to be found by multiplying the
number of stars nearer than r by the proportion with
linear velocities between rjr and v)(r + dr), for successive
steps of r. Thus if JVj and JV 2 be the total numbers of
stars with positive and negative components rj, we shall
have

and

J^-JH-^

so that

JTi+^i- {%)+!!( -*X-r FWg^Tjdr . . . (29)

%/ ""00

provided that we choose an even function to represent F(r).
We assume that the rj components are distributed accord
ing to the Maxwellian Law, so that

0(r)= * e-W-r? . (30)

VTT

Consider now the special form :

where k is a disposable constant.
Substituting in (29) and setting

rj = hi;
T = hV,

the equation reduces to

say.

The quantity T can be found from the ratio of the
number of negative proper motions 17 to the whole number.
This ratio is, in fact,

i j"

-

S'TT J r

22 4

STELLAR MOVEMENTS

CHAP.

Thus the function R(ri) for the particular region can be
tabulated. If then we can find a quantity k such that
the proportion of stars having component proper motions
numerically less than nk is represented by JR(n), then the
distance distribution is given by (31).

The form of f here discussed has been selected (by
F. W. Dyson and the writer independently) from other
forms lending themselves to mathematical treatment, as
corresponding most nearly to the observed motions. We
may make a second approximation by superposing two
such functions with different values of k.

Turning now to the results, we give first the distribu-
tion of stars brighter than 5 m '8 (Harvard Scale) for Types
A and K separately. It has been pointed out previously
that the assumed law (30) is only valid when stars of
homogeneous type are discussed. Two regions were
investigated, one with centre at the equinoxes (Gal.
Lat. 63), the other with centre at the poles (Gal. Lat. 27) ;
the former was much the more favourable for the purpose.

TABLE 37.
Distribution of Stars brighter them 5 m '8.

Limits of Distance.

Type K.

Type A.

Parsecs.

G. Lat. 63.

G. Lat. 27.

G. Lat. 63.

G. Lat. 27.*

25

11-7

9-2

7'9

7'5

25 50

30-7

24-1

' 18-2

21-4

50 75

35-3

31-5

18-8

327

75-] 00

31-4

31-8

17-6

38-5

100125

24-8

31-0

167

40-9

125-150

19-0

287

iu :; 39-0

150175

14-5

26-5

16-0

34-5

175200

11-3

23-3

15-0

27-9

L'OO >.",

8-2

ID'S

13-2

21-3

>r> 2r>0

57

15-6

11-2

15-3

>250

8-4

36-3

32-6

25-0

* North Polar area only. There appeared to be some abnormality near
the South Pole.

x STATISTICAL INVESTIGATIONS 225

Each region occupies 0*235 of the whole sky, and the
figures give the actual number of stars.

The results for Type K, which appear to be the more
trustworthy, show very well the falling off in numbers at
great distances in high galactic latitudes as compared with
low. The thinning out begins to be perceptible beyond
100 parsecs, and at 250 parsecs the density is only about
one-quarter of its value in the lower latitude. The
decrease is more rapid than would be expected from
Seeliger's figures. The results for Type A are not so easy
to accept. I am inclined to believe, however, that there
must be a real clustering of Type A stars in the low region
at a distance of about 100 parsecs, as the figures
suggest. Perhaps the cluster has special properties as
regards motion, which will upset the details of our table.
In the region of high latitude (for which there seems no
good reason to doubt the results) the stars appear much more
spread out than Type K, as would be expected if Type
A is rarer in space but of greater average luminosity. It
seems possible that the unexpectedly low average distance
of Type A (see p. 168) may be due to the presence of
extensive clusters of these stars near the Sun. It was
remarked that the next preceding type, B, shows a
great tendency to form moving clusters, and possibly
the same phenomenon in a vaguer form may remain in
Type A.

For the distribution of the fainter stars the proper
motions of 3,735 stars within 9 of the North Pole con-
tained in Carrington's Circumpolar Catalogue are available.
The limiting magnitude, which is the same as that of the
Bonn Durchmusteruug, is about 10 m '3. These stars have
been investigated by F. W. Dyson by a method similar
in principle to that we have just described. Dyson used
two forms of / (>), viz.

(a) /(r) = re-* 8 **
(6) /(r) = 7-0-s e -

226

STELLAR MOVEMENTS

CHAP.

The differences in the resulting distributions are small ;
the former probably gives the closer representation of the
nearer stars, and the latter of the more distant stars.
Both results are given in Table 38.

TABLE 38.
Distribution of Stars brighter than 10 m> 3 (for Galactic Latitude 27).

Limits of Distance.

Percentage of Stars.

Parsecs.

Formula (a).

Formula (6).

0- 40

0-9

1-0

40- 100

5-0

4-7

100 200

15'1

13-2

200 400

40'1

35-2

400 667

31-5

32-9

667-1000

7-1

11-9

>1000

0-3

1-1

As might be expected, these stars are considerably more
distant than those of Boss's Catalogue (cf. Table 37).
For example, only 57 per cent, are within 100 parsecs of
the Sun, as compared with more than 40 per cent, of the
Boss stars.

It may also be noted that 70 per cent, of the whole
number have parallaxes between 0"*005 and 0"*0015 ; or, in
terms of distance, 70 per cent are distant between 200
and 650 parsecs.

It is believed that in low galactic latitudes the actual
density of the stars in space is fairly constant up to a very
considerable distance from the Sun. On this hypothesis
Tables 37 and 38 enable us to determine the luminosity
law ; for the limiting magnitude of the stars considered
corresponds to a limiting absolute luminosity which
decreases as the square of the distance. If in the foregoing
investigations

STATISTICAL INVESTIGATIONS

227

/() dr is the number of stars between the limits of distance r to r+dr

in an area of the sky o>,

in is the limiting magnitude of the catalogue,
Q^'o is the Sun's stellar magnitude if removed to a distance of 1 parsec,

then the limiting luminosity for stars at a distance r is

(2 -512) 0-5 -m x r-',

and the number of stars per cubic parsec with luminosities
greater than this is

&).

or/ - ~

In Table 39 are given the luminosity laws derived from
the Boss proper motions (Type K only) and the Carrington
proper motions (all types). Owing to the difference in
brightness of the stars used, the two investigations apply
mainly to different parts of the luminosity curve.

TABLE 39.

Number of Stars in 4'2 x 10 6 units of space equal to a sphere of radius

100 parsecs.

Boss Stars (Type K only). Carrington Stars (All Types).

Luminosity Number Luminosity
(8un=l). of Stars. (Sun = l).

Number
of Stars.

>500 10-8

>200

400 to 500 7-3 | 100 to 200

24

300 400 12-9

50 100

316

200 300 26-3

25 50

1,190

150 200 25-0

10 25

3,310

100 150

51-9

1 10

18,360

50 100

139-0

o-i i

70,100

25 50

251-9

10 25

500

1 10

2,700

Total brighter than

Total brighter than

the Sun ....

3,725

the Sun ....

23,200

The agreement is not particularly good. We should
expect the numbers for Type K only to be considerably

Q 2

228 STELLAR MOVEMENTS CHAP.

less than for all the types taken together, so that for
luminosities less than 100 there is no noteworthy discord-
ance. For the Carrington stars, the number with
luminosity greater than 100 depends on stars distant
more than 900 parsecs ; as these are not more than 2 per
cent, of the whole, it is doubtful if the formulae ought to
be pressed so far. Also it may well be that at so great a
distance there is a real diminution of the density of dis-
tribution of the stars in space.

It is interesting to note that more than 95 per cent, of the
stars of Carrington (or of the Bonn Durchmusterung), and
nearly 99 per cent, of the stars brighter than 5 m *8, are
more luminous than the Sun. For a star of brightness
equal to the Sun to appear as bright as 10 m< 3 or 5 m> 8 it
must be distant less than 91 or 11 parsecs, respectively.
The percentages given are the proportions outside these
limits.

Adopting the law of distance-distribution for stars down
to a limiting magnitude

we may investigate the velocity-law in the direction of
star-streaming. This has been done by Dyson for the
Carrington proper motions. He found the best general
agreement was given by Schwarzschild's ellipsoidal hypo-
thesis rather than by the two-drift hypothesis. The ratio
of the two axes of the velocity ellipse was found to be
0*60, in close agreement with the investigations of much
brighter stars by the methods of Chapter VII. The two-
drift hypothesis does not represent the small proper
motions at all closely, though the existence of two maxima
in the observed distribution of motions lends support
to the view that there is some sort of separation of
the two streams. The disagreement of the small proper
motions is perhaps not surprising, for it is just this defect
'hat the third Drift was introduced to remedy ; but it is
certainly to the credit of the ellipsoidal hypothesis that

x STATISTICAL INVESTIGATIONS 229

with so few constants it can lead to a good agreement with
the small and large motions simultaneously.

The main points of discrimination between the two-
drift and ellipsoidal hypotheses appear to be

(1) The skewness of the velocity-distribution.

(2) The spread of the distribution.

(3) The existence (or not) of two maxima.

In the first point the advantage is admittedly with the
two-drift hypothesis, and it seems equally certain that the
ellipsoidal hypothesis is better fitted to the second. With
regard to (3) the evidence is rather in favour of the two
drifts. According to the relative importance attached to
these three criteria, different views of the merits of the
two methods of approximation will prevail.

One point of great practical importance must be at-
tended to in investigations of the kind we have been
considering (Section III.). The distribution of the proper
motions is generally appreciably altered by the presence
of accidental error in the observations. Usually the pro-
bable error of the observations is known, and in this case
we can, from a table of the number of observed proper
motions between given limits, form a revised table giving
the true number, i.e., we can correct the observed values
of h(rf) for the effects of the known accidental error. 6
Unless the accidental error is large, the correction to be
applied to each number of the observed table is

_/r046x probable errory x ^^ ^^ difference
\ tabular interval /

18-

The full formula, applicable to any kind of statistics,

-> -(-)**

where v (m) and u (m) are the true and observed frequency
functions, and 0'477/A is the probable error of the observa-
tions of m.-

230 STELLAR MOVEMENTS CHAP.

GENERAL CONCLUSION.

The investigations described in this chapter suffer
from all the imperfections inseparable from pioneer
work. The numerical results in Sections I. and II.
would, I believe, be modified in an important degree,
by a modern determination of the mean parallaxes of
stars of different magnitudes ; those of Section III.
suffer from scantiness of data, insufficient range, and the
absence of an effective check on the main assumptions.
Yet I do not think it can be doubted that these general
statistical researches have already greatly advanced our
knowledge of the distribution and luminosities of the
stars. If the approximation is not yet a close one, our
present vague knowledge is very different from the
complete ignorance from which we started. But the main
interest of this chapter lies in the hope for the future.

It is of special importance to note that there exist two
entirely independent methods of determining the distribu-
tion in distance of stars brighter than a limiting magnitude,
the one resting on magnitude-counts and mean parallactic
motions (Section I.),* and the other on the distribu-
tion of the individual proper motions (Section III.).
There is nothing in common between the data used for
these two methods ; and the one is a complete check
on the other. When the time arrives that this check
is satisfied, and that results obtained along one line of
investigation are in full agreement with those obtained
along an independent line, the results of these methods of
research will have been placed on a firm basis. Meanwhile,
the conclusion that such a check is possible may be
regarded as one of the most useful results of these
preliminary discussions.

* The method of Section I. leads, as we have seen, to a complete solution
for the luminosity-law and density-law, The distribution in distance of stars
brighter than a limited magnitude is easily derived from these.

x STATISTICAL INVESTIGATIONS 231

REFERENCES. CHAPTER X.

1. Charlier, Lund Meddelanden, Series 2, No. 8, p. 48.

2. Eddington, Monthly Notices, Vol. 72, p. 384.

3. Comstock, Astron. Jour., No. 655.

4. Seeliger, Sitzungsberichte, K. Bayer. Akad. zu Munchen, 1912, 'p. 451.

5. Kapteyn. Astron. Journ., No. 566, p. 119.

6. Eddington, Monthly Notices, Vol. 73, p. 359.

BIBLIOGRAPHY.

SECTIONS I. AND II. Kapteyn's principal investigations are :
Groningen Publications, No. 8 (1901) Mean parallaxes.

,, ,, No. 11 (1902) Luminosity and density laws,

with further developments and revision in
Astron. Journ., No. 566 (1904).

Proc. Amsterdam Acad. Sci., Vol. 10, p. 626 (1908).

Seeliger's investigations of the luminosity and density laws are contained
mainly in four papers.

K. Bayer. Akad. der Wiss. in Munchen, Abhandlungen, Vol. 19, Pt. 3
(1898), and Vol. 25, Pt. 3 (1909) ; ibid., Sitzungsberichte, 1911, p. 413, and
1912, p. 451.

The most important parts of the mathematical theory are given very
concisely by

Schwarzschild, Astr. Nach., Nos. 4422 and 4557.

SECTION III. The subject is treated by

Dyson, Monthly Notices, Vol. 73, pp. 334 and 402.

Eddington, Monthly Notices, Vol. 72, p. 368, and Vol. 73, p. 346.
Discussions by methods different in the main from those here described
are given by

Charlier, Lund Meddelanden, Series 2, Nos. 8 and 9.

v. d. Pahlen, Astr. Nach., No. 4725.

CHAPTER XI

THE MILKY WAY, STAR-CLUSTERS, AXD NEBULAE

AT the beginning of the preceding chapter we emphasised
the fact that the statistical investigations referred to an
idealised sidereal system, which retained some of the more
important properties of the actual universe, but neglected
many of the details of the distribution. If an impression
has been given that the spheroidal distribution of stars
with density diminishing outwards is a complete and
sufficient model, a glance at Plate I (Frontispiece),
which is a photograph of the region of the Milky Way in
the neighbourhood of Sagittarius, may serve to correct
this. In the Milky Way there are unmistakable signs of
clustering and irregularities of density on a large scale.
The great star-clouds and deep rifts are features in
marked contrast with the phenomena of distribution
that we have hitherto considered, and no elaboration of
the theory of a disc- shaped or spheroidal system will suf-
fice to explain them. This does not affect our conclusions
as to the shape of what we have called the inner stellar
system ; a general concentration of stars to the galactic
plane is manifested quite apart from the great clusters
of the Milky Way itself. It would certainly be desirable
in discussing problems such as those of the last chapter to
ignore, or at least treat separately, the parts of the sky
through which the Milky Way itself passes, for our
idealised system evidently becomes inadequate here.

CH. xi CLUSTERS AND NEBULA 233

The view then is taken that there is first an inner stellar
system consisting of a flattened distribution of stars of
density more or less uniform at the centre and diminishing
outwards ; and secondly a mass of star-clouds, arranged
round it and in its plane, which make up the Milky Way
(see Fig. 1, p. 31). It is to the inner system that our
knowledge of stellar motions and luminosity relates.
Whether the outer clouds are continuous with the inner
system or whether they are isolated, is a question at
present without answer.

It is usual to consider the system in the midst of which the
Sun lies as the principal system, the clusters of the Milky
Way being a kind of appanage. An alternative view makes
no such distinction, but contemplates a number of star-
clouds scattered irregularly in the one fundamental plane,
our own system being one of them. There are certain ad-
vantages in the latter view, especially as the two star-streams
could then be accounted for by two of these star-clouds
meeting and passing through one another. By a natural
reaction from the geocentric views of the Middle Ages, we
are averse to placing the earth at the hub of the stellar
universe, even though that distinction is shared by
thousands of other bodies. But it is doubtful if there is
really any close resemblance between the Milky Way
aggregations and that which surrounds the Sun. We do
not recognise in them the oblate form flattened in the
fundamental plane, which is so significant a feature of the
solar star-cloud. They seem to be of a more irregular
character and for that reason we prefer to adhere to a
theory which regards them as subsidiary.

The great mass of the stars shown in photographs of
the Milky Way are very faint. We have no knowledge
of their motions or their spectra, and even now there is
but little accurate information as to their magnitudes and
numbers. It is an important question whether some of
the bright -stars, which are seen in the same region as these

234 STELLAR MOVEMENTS CHAP.

star-clouds, are actually in the clouds or only projected
against them. The investigations of this point are rather
contradictory ; but on the whole it seems probable that
some stars of the sixth magnitude are actually situated in
the Milky Way clusters. Certainly by the ninth magnitude
we have begun to penetrate into the true galaxy, and the
twelfth or thirteenth magnitude takes us right into the
heart of the aggregations. Simon Newcomb l attacked the
problem by comparing the density of the lucid stars,
where the Milky Way background was respectively bright
and faint ; he found that bright stars were most numerous
where the background was bright.

TABLE 40.
Relation of Lucid Stars to the Milky Way (Newcomb).

N. Hemisphere
(Limiting Mag.
6'3).

S. Hemisphere
(Limiting Mag.
7-0).

Mean star-density of whole hemisphere
Star-density : darker galactic regions .
,, brighter ,, ,,

19-0
20-4
32-9

327
33-8
79-4

The star-density is given per 100 square degrees. The
greater value in the southern hemisphere is due to the
lower limit of magnitude in the catalogue which was
used.

The condensation of the stars to the brighter regions is
very marked ; but some caution is needed in interpreting
this result. There is no doubt that many of the dark
patches in the Milky Way are due to the absorption of
light by tracts of nebulous matter. To perform their
work of absorption these tracts must lie on the nearer side
of the Milky Way aggregations. Since Table 40 shows that
the star-density in the darker regions is barely greater than
the mean for the hemisphere, and therefore less than in a
zone just outside the Milky Way, the dark matter must be at

XI

CLUSTERS AND NEBULA

235

least partly within the oblate inner system. Newcomb's
result therefore teaches us that some of the sixth and seventh
magnitude stars lie in and beyond the dark clouds ; but it is
not conclusive that any of them lie in the bright aggrega-
tions of the Milky Way. To prove the latter result we
should have to show that the star-density in the bright
regions is greater than could reasonably be attributed to
the oblate shape of the inner system ; the figures suggest
that this is so, but there is room for doubt.

A similar discussion of fainter stars has been made by
C. Easton, 2 who considered especially the part of the Milky
Way in Cygnus and Aquila , where there is a wide range
in the intensity of the light. A selection from his results
is given in Table 41.

TABLE 41.
Relation of Stars to the Milky Way (Easton).

Argelander Wolf

W. Herschel

Durchmusterumj Photographs

Star-gauges

(Mag. 010).

(Mag. 011).

(Mag. 014).

Star-density

Darkest patches 23

72

405

Intermediate ,, 33

134

4114

Brightest ,, 48

217

6920

The density is given per square degree.

The numbers show that as we proceed to fainter stars a
rapidly increasing proportion is associated with the true
Milky Way aggregations, and by the time the fourteenth
magnitude is reached, an overwhelming proportion is
found to belong to them. But Easton's results and
Xewcomb's are not quite in agreement. The relative
superiority of the bright patches found by Easton for 10 m
is barely greater than that found by Newcomb for 6 m -7 m .
Indeed by extrapolation of Easton's figures we should
conclude that the stars brighter than 7 m were not notice-

236 STELLAR MOVEMENTS CHAP.

ably associated with the Milky Way background. If we
could suppose that the Cygnus-Aquila region is more
distant than the average, the difference between the two
results would be explained ; but Easton has given reasons
for believing that this region is nearer than the average.

In all the counts discussed by Easton. the star-density
of the bright patches is so notably superior to the density
just outside the Milky Way that we must conclude that
the excess is actually due to the star-clouds. This method
of considering the problem is so straightforward that it
seems impossible to doubt the conclusions. The results
should be independent of systematic errors in the limits
of the counts, for the bright and dark regions adjoin one
another and are irregularly intermixed. If there is any
tendency to make the counts less complete in the rich
regions or where the background is bright, this only
means that the difference is really more accentuated than
is shown in the Table.

If it is established that the Milky AVay aggregations
include a fair number of stars that appear to us of the
ninth magnitude, it is possible to form some conception
of their distance. A star of luminosity 10,000 times
that of the Sun would appear of the ninth magnitude at
a distance of 5,000 parsecs. This may be taken as an
upper limit for the distance of the nearer parts of the
Milky Way. If, following Newcomb, we admit the
presence of sixth magnitude stars in the aggregations, the
limit is reduced to 1200 parsecs. In any case the
so-called li holes " in the Milky Way (dark nebulse) would
seem to be in some cases within this latter distance.
There is no reason to believe that all parts of the galaxy
are at the same distance, and certain appearances suggest
that there may be two or more branches lying behind one
another in some parts of the sky. The relative bright-
ness of the different portions gives no clue to the distance ;
for the apparent brightness (per unit angular area) of a

;<

xi CLUSTERS AND NEBULA 237

cluster of stars is independent of its distance.^ The
differences in intensity must, therefore, be due either to a
greater depth in the line of sight or to a closer concen-
tration of the stars. (If there is appreciable general
absorption of light in space, this statement would have
to be modified.) Further, as it is necessary to suppose
that the whole structure is quite irregular, it would be
very unsafe to assume that the angular width of the
different portions gives any measure of the distance.

It is well known that the large irregular nebulae are
found principally in the Milky Way, differing in this
respect from the great majority of the compact nebulae.
In the "irregular " class we include not only the more intense
patches, such as the Omega, Keyhole, and Trifid nebulae,
etc., but the extended nebulous backgrounds, such as those
photographed by E. E. Barnard in Taurus, Scorpius, and
other constellations. Of the same nature are many of the
dark spaces in the Milky Way, where the light of the stars
behind is cut off by nebulous regions which give little or
no visible light of their own. These dark spaces are
usually connected with diffused visible nebulosity, which
often surrounds and is condensed about one or more bright
stars. An excellent example is found in the constellation
Corona Austrina (Plate 2), where there is a dark area con-
taining very few stars, edged in some parts with visible
nebulosity, which condenses into brig cusps round the
bright stars. No one examining < photographs can
doubt that the darkness is caused by the nebula the edge
of which is visible. Visual observers have asserted that
the region has a leaden or slightly tinted appearance as
though a cloud were covering part of the field. 3 Another
curious region is found in Sagittarius near the cluster
Messier 22, where great curved lanes are, as it were,
.smudged out among the thickly scattered stars. The

* Nearness might decrease the background illumination somewhat, for
a greater amount of the light would appear in the form of distinct stars.

2 4 o STELLAR MOVEMENTS CHAP.

association. In the Greater Cloud (Plate 3) there are a large
number of nebulous knots, which have generally been
described as spiral (i.e., non-gaseous) nebulae. If this were
really their nature it would form a remarkable distinction be-
tween the clouds and the Milky Way, for the spiral nebulae
avoid the latter. But according to A. R. Hinks the
supposed nebulae of the Magellanic Clouds are unlike any-
thing found elsewhere, and have no resemblance to the true
spiral nebulae. Many of the principal nebulae in the Cloud
are undoubtedly gaseous.

By an ingenious argument E. Hertzsprung 7 has arrived
at an estimate of the distance of the Lesser Magellanic
Cloud, which is entitled to some confidence. It depends
on the existence of a large number of variables of the
S Cephei type in that Cloud. Now there is reason to believe
that the absolute magnitude of a Cepheid variable of given
period is a fairly definite quantity ; that, in fact, it can be
predicted from the period with a mean uncertainty of
only a quarter of a m'agnitude. This was shown by Miss
Leavitt 8 who discussed the variables in the Lesser Cloud.
Since these must be at nearly the same distance from the
Sun, their apparent magnitudes will differ from their
absolute magnitudes by a constant. She found that the
magnitude and the logarithm of the period were connected
by a linear relation, and that the average deviation of any
individual from the general formula was m *27. Now
the mean distance of the brighter Cepheid Variables can
be calculated from the paral lactic motion in the usual way.
It is then only necessary to multiply by the factor for
the difference of magnitude between these and the Magel-
lanic Variables and allow for the difference of period, if
any, in order to obtain the distance of the latter. In this
way the distance of the Lesser Magellanic Cloud is found
to be 10,000 parsecs the greatest distance we have yet
had occasion to mention.

Passing from the large-scale aggregations of stars, we

Plate 3.

Franklin-Adams Chart.

THE GREATER MAGELLANIC CLOUD.

xi CLUSTERS AND NEBULA 241

must refer briefly to the star-clusters proper. There
seems no reason to doubt that these are of the same nature
as the moving clusters discussed in Chapter IV. In par-
ticular the Taurus-stream may be taken to be typical of
the globular clusters, although it is not one of the richer
specimens of the class. The distribution of these globular
clusters in the sky is very remarkable ; they are to be found
almost exclusively in one hemisphere of the sky, the pole of
which is in the galactic plane in galactic longitude 300.
This result (which is taken from A. R. Hinks's discussion 9 ) is
clearly of great significance ; but it does not seem possible
at present to attempt any explanation of it.

The chief characteristics, from our point of view, of the
planetary nebulae are their close condensation to the Milky
Way and the large radial velocities of those that have
been measured. Here again we are not able to do more
than state the facts. I am not aware of any trustworthy
measures of the proper motions of planetary nebulae, and
their size and distance are consequently a matter of extreme
uncertainty ; but their marked tendency to lie in the
plane of the Milky Way shows that they must be placed
somewhere within our own stellar system.

In Plate 4 is shown the Whirlpool Nebula in Canes
Venatici, a fine example of the spiral nebulas, which are
among the most beautiful objects in the heavens. It is
generally believed that the spirals predominate enormously
over the other classes of nebulae ; and, as the whole number
of nebulae bright enough to be photographed has been
estimated by E. A. Fath at 160,000, they must form a
very numerous class of objects. They are seen by us at
all inclinations, some, like the Whirlpool Nebula, in full
front view, whilst others are edge-on to us and appear as
little more than a narrow line. An example of the latter
kind is also illustrated in Plate IV. In all cases, where it is
possible to discriminate the details, the spiral is seen to
be double-branched, the two arms leaving the nucleus at

R

242 STELLAR MOVEMENTS CHAP.

opposite points and coiling round in the same sense.
From the researches of E. v. d. Pahlen 10 it appears that
the standard form is a logarithmic spiral. The arms, how-
ever, often present irregularities, and numerous knots and
variations of brightness occur. Unlike the planetary and
extended nebulae, the spectrum shows a strong continuous
background ; bright lines and bands are believed to occur,
at least in the Great Andromeda Nebula ; but they are of
the character of those found in some of the early type
stars, and are distinct from the emission lines of the
gaseous nebulae.

The distribution of spiral nebulae presents one quite
unique feature : they actually shun the galactic regions
and preponderate in the neighbourhood of the galactic
poles. The north galactic pole seems to be a more
favoured region than the south. This avoidance of the
Milky Way is not absolute ; but it represents a very
strong tendency.'

In the days before the spectroscope had enabled us to
discriminate between different kinds of nebulae, when all
classes were looked upon as unresolved star-clusters, the
opinion was widely held that these nebulae were " island
universes," separated from our own stellar system by a
vast empty space. It is now known that the irregular
gaseous nebulae, such as that of Orion, are intimately
related with the stars, and belong to our own system ;
but the hypothesis has recently been revived so far
as regards the spiral nebulae. Although the same term
" nebula " is used to denote the three classes irregular,
planetary, and spiral we must not be misled into suppos-
ing that there is any close relation between these objects.
All the evidence points to a wide distinction between
them. We have no reason to believe that the arguments
which convince us that the irregular arid planetary nebulae
are within the stellar system apply to the spirals.

It must be admitted that direct evidence is entirely

xi CLUSTERS AND NEBULAE 243

lacking as to whether these bodies are within or without
the stellar system. Their distribution, so different from
that of .all other objects, may be considered to show that
they have no unity with the rest ; but there are other
bodies, the stars of Type M for instance, which remain
indifferent to galactic influence. Indeed, the mere fact
that spiral nebulae shun the galaxy may indicate that they
are influenced by it. The alternative view is that, lying
altogether outside our system, those that happen to be in
low galactic latitudes are blotted out by great tracts of
absorbing matter similar to those which form the dark
spaces of the Milky Way.

If the spiral nebulae are within the stellar system, we
have no notion what their nature may be. That hypothesis
leads to a full stop. It is true that according to one
theory the solar system was evolved from a spiral nebula,
but the term is here used only by a remote analogy with
such objects as those depicted in the Plate. The spirals to
which we are referring are, at any rate, too vast to give birth
to a solar system, nor could they arise from the disruptive
approach of two stars ; we must at least credit them as
capable of generating a star cluster.

If, however, it is assumed that these nebulae are external
to the stellar system, that they are in fact systems coequal
with our own, we have at least an hypothesis which can be
followed up, and may throw some light on the problems that
have been before us. For this reason the " island universe "
theory is much to be preferred as a working hypothesis ;
and its consequences are so helpful as to suggest a dis-
tinct probability of its truth.

If each spiral nebula is a stellar system, it follows that
our own system is a spiral nebula. The oblate inner
system of stars may be identified with the nucleus of the
nebula, and the star clouds of the Milky Way form its
spiral arms. There is one nebula seen edgewise (Plate IV)
which m'akes an excellent model of our system, for the

R 2

244 STELLAR MOVEMENTS CHAP.

oblate shape of the central portion is well-shown. From
the distribution of the Wolf-Ray et stars and Cepheid
Variables, believed to belong to the more distant parts of
the system, we infer that the outer whorls of our system
lie closely confined to the galactic plane ; in the nebula
these outer parts are seen in section as a narrow rectilinear
streak. The photograph also shows a remarkable absorp-
tion of the light of the oblate nucleus, where it is crossed
by the spiral arms. We have seen that the Milky Way
contains dark patches of absorbing matter, which would
give exactly this effect. Moreover, quite apart from the
present theory, a spiral form of the Milky Way has been
advocated. Probably there is more than one way of repre-
senting its structure by means of a double-armed spiral ;
but as an example the discussion of C. Easton n may be
taken, which renders a very detailed explanation of the
appearance. His scheme disagrees with our hypothesis
iu one respect, for he has placed the Sun well outside the
central nucleus, which is situated according to his view in
the rich galactic region of Cygnus.

The two arms of the spiral have an interesting meaning
for us in connection with stellar movements. The form
of the arms a logarithmic spiral has not as yet given
any clue to the dynamics of spiral nebulae. But though
we do not understand the cause, we see that there is a
widespread law compelling matter to flow in these forms.

It is clear too that either matter is flowing into the
nucleus from the spiral branches or it is flowing out from
the nucleus into the branches. It does not at present con-,
cern us in which direction the evolution is proceeding. In
either case we have currents of matter in opposite direc-
tions at the points where the arms merge in the central
aggregation. These currents must continue through the
centre, for, as will be shown in the next chapter, the
stars do not interfere with one another's paths. Here
then we have an explanation of the prevalence of motions

xi CLUSTERS AND NEBULAE 245

to and fro in a particular straight line ; it is the line from
which the spiral branches start out. The two star-
streams and the double-branched spirals arise from the
same cause.

REFERENCES.

1. Newcomb, The Stars, p. 269.

2. Easton, Proc. Amsterdam Acad. Sci., Vol. 8, No. 3 (1903); Astr.
Nach., Nos. 3270, 3803.

3. Knox Shaw, The Observatory, Vol. 37, p. 101.

4. W. Herschel, Collected Papers, Vol. 1, p. 164.

5. Hertzsprung, Astr. Nach., No. 4600. See a Iso Newcomb "Contribu-
tions to Stellar Statistics, No. 1" (Carnegie List. Pub., No. 10.)

6. Harv. Ann., Vol. 56, No. 6 ; Newcomb, The Stars, p. 256.

7. Hertzsprung, Astr. Nach., No. 4692.

8. Leavitt, Harvard Circular, No. 173.

9. Hinks, Monthly Notices, Vol. 71, p. 697.

10. v. d. Pahlen, Astr. Nach., No. 4503.

11. Easton, Astrophysical Journal, Vol. 37, p. 105.

-

CHAPTER XII

DYNAMICS OF THE STELLAR SYSTEM

DURING the time that the stars have been under
observation their motion has been sensibly rectilinear
and uniform. A reservation must be made in the case of
binary stars, where the components revolve around one
another ; but from the present point of view pairs or
multiple systems of this kind only count as single
individuals. With this exception, we have no direct
evidence that one star influences the motion of another ;
yet we cannot doubt that, in the vast period of time
during which the stellar universe has been developing, the
forces of gravitation must have played a part in shaping
the motions that now exist. It may not be premature to
consider the dynamical aspects of some of the discoveries
of recent years.

The action of one star on another, even at the smallest
normal stellar distance is exceedingly minute. The attrac-
tion of the Sun on a Centauri imparts to that star in the
course of a year a velocity of one centimetre per hour. At
this rate it would take 380,000,000 years to communicate
a velocity of one kilometre per second. The period is not
so excessive, compared with what we believe to be the
life of a star, as to entitle us to despise such a force. But
the two stars will not remain neighbours for more than a
small fraction of that time. Although a Centauri is at
present approaching, a separation will soon begin ; in

CH. xii DYNAMICAL THEORY 247

150,000 years from now the distance will have doubled;
and before the communicated velocity amounts to more
than a fraction of a metre per second the star will have
receded out of range of the Sun's attraction.

The case is different when we consider the general
attraction of the whole stellar system on its members.
Not only is the magnitude of the force somewhat greater,
but the time through which its effects accumulate is far
longer than in the case of one star acting on a temporary
neighbour. This general attraction is quite sufficient to
produce important effects on the stellar movements.

The field of force in which a star moves is due to
a great number of point-centres the stars. The distribu-
tion of the attracting matter is discontinuous. We there-
fore divide the force into two parts, (1) the attraction
in an ideal continuous medium having the same average
density, and the same large-scale variations of density
as the stellar system, and (2) the force due to the
accidental arrangement of the stars in the immediate
neighbourhood. The same distinction occurs in the
ordinary theory of attractions for points inside the
gravitating matter. AVe call the first part the general
or central attraction of the system ; the term central is
perhaps inaccurate for there is no true centre of attraction
unless the system possesses spherical symmetry. The
second part is of an accidental character and will act in
different directions at different times ; but it is not on
that account to be dismissed without consideration.

The stars have often been compared to the molecules of
a gas ; and it has been proposed to apply the theory of
gases to the stellar system. 1 The essential feature in gas-
dynamics is the prominent part taken by the collisions of
the molecules. Now it is clear that collisions of the stars,
if they occur at all, must be exceedingly rare ; and the
effect would certainly not be the harmless rebound
contemplated by the theory of gases. It is, however,

248 STELLAR MOVEMENTS CHAP.

well-known that the precise mode of interaction during the
encounter is of little importance, and all that is required
in the theory is an interchange of momentum taking place
between two individuals in their line of centres. In this
generalised sense encounters are continually taking place ;
the passage of one star past another always involves some
interchange of momentum. It remains to be examined
whether this continuous transference can play the same
part in stellar theory as the abrupt changes of momentum
in the gas theory.

In the long run the same effect will be brought about.
The ultimate state of a system of gravitating stars will be
the same as that of a gas. The ultimate law of velocities
will be the same as in a mass of non-radiating monatomic
gas under its own attraction ; and moreover there will be
equipartition of energy between the stars of different
masses, just as if they were atoms of different weights.
We might even go further , and look forward to a still
more " ultimate " state, in which the double stars behaved
as diatomic molecules. But it is unnecessary to pursue
these deductions, for they have no reference to anything
in the present state of the stellar universe, or to any
future near enough to interest us.

It was seen in Chapter IV. that the existence of the
Moving Clusters shows plainly that the encounters have
not as yet had any appreciable effect on the motions of the
stars. Taking, for example, the Taurus Cluster, we have
seen that it occupies a sphere of about 5 parsecs radius,
which would in an ordinary way contain 30 stars. As it can-
not be supposed that a special track has been cleared for
the passage of the cluster, the stars th;it would naturally
occupy the space must be there, permeating the cluster
without belonging to it. hi so far as they have any
effect at all, the attractions of these interlopers must tend
to break up and dissipate the cluster, by destroying the
parallelism of the motions. As no such breaking up has

XII

DYNAMICAL THEORY

249

taken place, it may be inferred that the chance encounters
have had no appreciable effect on the stellar velocities up
to the present time. Many of the stars of the Taurus
Cluster are in a mature stage of development, so that this
inference may fairly be applied to the general mass of
the stars.

A consideration of this question from the theoretical side
is in entire accordance with this conclusion. We begin by
considering the numerical amount of the deflection pro-
duced by an encounter in given circumstances.

Fig. 22.

Let S l S. 2 (Fig. 22) be two stars of masses m l and m 2 and
G be their common centre of gravity.
Since

we may replace the star S. 2 , in considering its attraction on
S lt by a star of mass

ml m * Yat.

"A?^ + JI1 2 /

Taking G to be at rest, let the star S L move along the
hyperbolical path HAH f starting with an initial velocity
V. Let CH, CH f be the asymptotes of the hyperbola.

Draw G Y perpendicular to CH ; then G Y is equal to
the transverse axis b of the hyperbola.

2 5 o STELLAR MOVEMENTS CHAP

The usual equation h 2 = pi gives in this case

Hence

a = -^s
The deflection i|r =

180- HCH'
is given by

or, since ty is always a small angle,

, 2^
6F 2 '

The transverse velocity imparted by the encounter is

If U is the initial relative velocity of the two stars,
(T the distance of nearest approach calculated as if no
deflection were to take place,

F = 2 _ u,

w*! + w 2
h W 2

m l + m 2

Also

m- \ 2
? ) ,

l + m 2 /

where 7 is the constant of gravitation.
Hence the transverse velocity imparted is

It may be noted that this expression does not involve
m lt so that the tendency towards equipartition of energy
is not indicated in the formula. Equipartition appears to
be a third-order effect depending on >|r 3 , which has been
neglected in the foregoing analysis.

Close approaches which produce an appreciable deflection

xii DYNAMICAL THEORY 251

-fy- are exceedingly rare. Evidently the probability of such
an event happening can be calculated when the density of
stellar distribution is known. It is of greater importance
to determine what is the cumulative effect of the large
number of infinitesimal encounters experienced by a star
in the course of a long period of time. The following
discussion is based on an investigation by J. H. Jeans. 2

We may set two limits, CT O and ov The former is the
upper limit of distance for sharp encounters producing
considerable deflections ; these will be treated as excep-
tional events to be studied separately. The latter is an
arbitrary limit beyond which approaches will not be held
to constitute an encounter.

Let v be the number of stars per unit volume.

The mean free-path (as defined by Maxwell) is

i

V27T./0-! 2 '

Thus in a length of path L the total number of en-
counters to be expected is

N = 2*m,<r*L.

The transverse velocity imparted at any encounter has
been shown to be

2ym. 2
U<r'

The average value of the relative velocity U is some-
what greater than the velocity v of the star S l relative to
the stellar system, for collisions will most frequently
occur with those stars which are coming to meet S lf

The, average value of I/a-

= I - 2n<r da- -f- I 2rro- d<r

J o " J o

2/<r r

As each encounter takes place in a haphazard direction,
the individual contributions of transverse velocity must
be compounded according to the theory of errors. Thus
the probable resultant of N encounters is proportional

252 STELLAR MOVEMENTS CHAP.

to ^/N ; and, generally, the square of the probable re-
sultant will be the sum of the squares of the indivi-
dual deflections. Thus in averaging different sorts of
encounters, we ought to use the mean- square values.

The mean-square value of is somewhat greater than

cr

2

.* We may conveniently regard this excess as roughly
a i

cancelling the excess of U over v and write the resultant
transverse velocity after N encounters

and the resultant deflection (in radians)

Substituting the value of N, the deflection becomes

The following numerical results are deduced by Jeans,
on the assumption that the density of stellar distribution
is one star to a sphere of radius one parsec, and that the
average mass of a star is five times that of the solar

* As the mean-square value appears at tirst sight to be infinite some
further explanation may be desired. The ratio of the mean-square value of

- to the simple mean is

<T

which is analytically infinite.

If, however, we reserve sharp encounters for separate consideration wo
may set <r () instead of for the lower limit. For a deflection of 2, <r will be
about 500 astronomical units, o-j need certainly not be taken greater than
500,000,000 astronomical units, for an i ncounter at that distance would last
an indefinite time. With tln-M- values

Thus the mean-s<|uarc \alue is nut more than 1M'> times the simple mean,
sudden deflections of over '1 In-ill^ reserved.

xii DYNAMICAL THEORY 253

system. This density is somewhat greater than that which
we have regarded as probable, and, accordingly, the results
may exaggerate a little the disturbance due to encounters.

Jeans finds that for an average star a deflection of 1
may be expected after 3,200 million years In addition to
this there is a small " expectation " of deflection by violent
encounters. From such a cause a deflection of 2 or more
might be expected once in a period of 8 x 10 11 years. The
meaning of these figures may be illustrated by a definite
instance. Considering a moving cluster, all the stars of
which have equal and parallel velocities of 40 km. per sec.,
let a star be considered to continue a member of the main
stream so long as its direction of motion does not diverge
by more than 2. After 100 million years only 1 in 8,000
of the original members will be lost by violent encounters,
and the remainder will make angles with the main stream
of which the average amount is only 10'. After 3,200 million
years the loss will be 1 in 250, and the average angle of the
remainder will be 1. After 80,000 million years one-tenth
of the original members have been lost by sharp encounters,
but the average angle of the remainder is 5. The ultimate
dissolution thus takes place mainly by gradual scattering
and not by sharp encounters.

It must riot be overlooked that a cluster possesses a
certain cohesion of its own, which may resist the minute
scattering force to which it is subjected. The cluster is a
place where the stars are grouped more densely than in
the rest of space and a gravitational force is exerted on
those which tend to stray away from it. As we cannot
even roughly estimate the whole number of stars in any of
the clusters, it is not possible to determine the amount of
this force ; but a simple calculation will show that it must
play some part in keeping the cluster together. Taking
Boss's Taurus-stream, which moves at the rate of 40 km.
per second, it has been stated that a deflection of 1 may
be expected after 3,200 million years, which is equivalent

STELLAR MOVEMENTS CHAP.

to 1' after a million years. This deflection is equivalent
to a transverse velocity of 0*012 parsec per million years.
It is easily shown that the probable transverse displace-

f . Ar - ir . n i 0-012 x

ment after Jy million years will be - - - - parsecs.

At the present time the stars of this cluster have spread
away from the mean position by an average distance of
about 3 parsecs. The corresponding value of 2V is 57.
This calculation gives 57 million years as an upper limit
of the age of the Taurus Cluster, assuming that its present
extension is wholly due to encounters. The result
depends on the rather high value of the stellar density
used by Jeans, but with a much smaller density the period
is still unreasonably short. It is clear therefore that the
dissolving effect of the encounters has been very largely
counteracted, and presumably the opposing circumstance is
the mutual gravitation of the members of the cluster.

This consideration does not destroy the force of our
previous argument. The cohesion of the cluster is only
important because the dissolving forces are so excessively
minute. Jeans's calculation applies directly to an inde-
pendent star, and shows that it can pursue its course
practically unmolested ; whilst the observational evidence
is that the effect of encounters is so small that even the
minute attraction in a moving cluster is sufficient to
counteract it. The evidence, observational and theoretical,
seems so conclusive that we have no hesitation in accepting
it as the basis of stellar dynamics. The apparent analogy
with the kinetic theory of gases is rejected altogether, and
it is taken as a fundamental principle that the stars
describe paths under the general attraction of the stellar
system without interfering with one another.

Let us now attempt to estimate the order of magnitude
of the general attraction, or the time required for a star
to describe its orbit. If the stellar system is not spheri-
cally symmetrical the orbits will not in general be closed

xii DYNAMICAL THEORY 255

paths. But for our estimates we need no precise definition
of the periodic time ; we want to know roughly how long
it takes for a star to pass from one side to the other of the
sidereal system and back again. Within a sphere of
uniform density all stars would describe elliptic orbits
about the centre isochronously, whatever the initial con-
ditions. The period depends only on the density and
is independent of the size of the sphere. The greater
the density the less will be the period according to the
relation T oc p~*. In an ellipsoidal system, the component
motions along the principal axes will be simple harmonic
but with differing periods ; the period in a spherical
system of the same density will be intermediate between
them. Thus the period calculated simply from the density
on the assumption of a spherical distribution will give
a general idea of the period in the actual universe.

\Yc have estimated the number of stars in a sphere of
radius 5 parsecs to be 30 ; as these stars are mostly fainter
than the Sun, we shall take the mass in this sphere to be
only 10 times the Sun's. If this is an underestimate, and
it may be very much underestimated on account of the
possible presence of dark stars, the period obtained will
be an upper limit. With the adopted density the result
is 300,000,000 years. This is less than current estimates
of the age of the solid crust of the earth. Thus the Sun
and other stars of like maturity must have described at
least one and probably many circuits, since they came
into being. * We are justified in thinking of the stellar
orbits as paths that have actually been traversed, and
not as mere theoretical curves.

The problems on which dynamics would be expected
to throw some light are numerous. Why have the stars

* We have no means of estimating the age of the stellar system, but
perhaps there is no harm in having some such figure as 10 10 years at the
back of our minds in thinking of these questions.

256 STELLAR MOVEMENTS CHAP.

in the early stages very small velocities ? Why do these
velocities afterwards increase ? In particular, how do the
stars acquire the velocities at right angles to the original
plane of distribution, which cause the latest types to be
distributed in a nearly spherical form ? How are the two
star-streams to be explained ? What is the meaning of
the third stream, Drift ? Can the partial conforming
to Maxwell's law be accounted for ? What prevents the
collapse of the Milky Way ?

Some of these problems seem to be at present quite
insoluble. Indeed, it must be admitted that very little
progress has been made in the application of dynamics
to stellar problems. What has been accomplished is
rather of the nature of preparatory work. It has been
shown that stellar dynamics is a different study from gas-
dynamics, and, indeed, from the theory of any type of
system that has yet been investigated. A regular pro-
gression may be traced through rigid dynamics, hydro-
dynamics, gas-dynamics to stellar dynamics. In the first
all the particles move in a connected manner; in the
second there is continuity between the motions of con-
tiguous particles ; in the third the adjacent particles act
on one another by collision, so that, although there is no
mathematical continuity, a kind of physical continuity
remains ; in the last the adjacent particles are entirely
independent. A new type of dynamical system has
therefore to be considered, and it is probably necessary
first to work out the results in simple cases and to become
familiar with the general properties, before attempting to
solve the complex problems which the actual stellar
universe presents. This has been the mode of develop-
ment in the other branches of dynamics.

The natural starting point is to investigate the possible
steady states of motion. It will be understood that we
are not here referring to the ultimate steady state in which
the gas-distribution of velocities prevails, but a state which

xii DYNAMICAL THEORY 257

remains steady so long as the effect of the encounters is
negligible. The actual stellar system may or may not be
in such a state ; we can best hope to settle the question by
working out the consequences of making that hypothesis.

For systems possessing globular symmetry a number of
types of steady motion have been found and investigated. 3
None of those discovered up to the present give a reasonable
approximation to the actual distribution of velocities ; but
the failures seem to narrow down very considerably the
field in which possible solutions may be obtained. A self-
consistent dynamical model, possessing some at least of
the main features of stellar motions, would be a most use-
ful and suggestive adjunct in many kinds of investigation,
and it would seem to be well worth while continuing the
search for suitable systems.

There is one problem relating to the actual stellar
system which may be kept in view even at this early
stage. H. H. Turner 4 has made an interesting suggestion,
which gives a possible explanation of the two star-streams.
The problem is to account for preferential motion to and
fro in one particular line. Suppose that the stars move in
orbits which are in general very elongated, somewhat like
the cometary orbits in the solar system. The stellar
motions then will be preferentially radial rather than
transverse. If the Sun is at a considerable distance from
the centre of the stellar system, the line joining that
centre to the Sun will be a direction of preferential motion
for those stars which are sufficiently near us to have
sensible proper motion. Even if the eccentricity of the
Sun is not very great, an effect of the character of star-
streaming will be observed. We have always assumed
that the convergence of the apparent directions of star-
streaming in different parts of the sky was evidence that the
true directions were parallel ; but a convergence of the
true directions is an equally possible interpretation. It is
quite possible that the preferential motions may be towards

258 STELLAR MOVEMENTS CHAP.

or away from a point at a finite distance rather than
parallel to a line. It is difficult to say whether such a
hypothesis would prove satisfactory in detail ; but at least
there is no obvious objection to it.

It may be asked, Why should the stellar orbits be very
elongated ? The reason that may be assigned is that they
start initially with very small velocities. We must sup-
pose that their velocities in later stages are mainly acquired
by falling towards the centre of the system, and it is only
natural that they should be preferentially radial. Indeed,
surprise has sometimes been expressed that there is so little
indication of the preponderating radial movements that
might be expected. Is it possible that the perplexing
phenomenon of the two star-streams is just this indication?

The great difficulty is that, if the motions are mainly
radial, it seems inevitable that there should be a great
congestion of stars in the neighbourhood of the centre-
greater than we care to accept as possible. In the systems
that have been worked out up to the present, it has nob
been possible to obtain sufficient preferential motion with-
out too great a density at the centre ; but we cannot as yet
conclude that this holds generally. Of course, it is possible
to argue that one of the dense patches of stars either in the
Cygnus or Sagittarius regions of the Milky Way is the
actually congested centre of the stellar system.

A few remarks may be made on the other problems
suggested on p. 256. The birth of a star without motion
does not seem to present so much difficulty as has some-
times been supposed. It is not necessary to suppose that
the primordial matter from which it arises is not subject to
gravitation (though there is nothing inherently improbable
in such a speculation). Presumably a star is formed by the
gathering together of meteoric or gaseous material in some
portion of space. Now, we know that, if we were to lump
together a thousand stars, their individual motions would

xii DYNAMICAL THEORY 259

practically cancel and the resultant super-star would be
nearly at rest. Similarly, in forming a single star, the
individual motions produced by gravitation in the materials
of which it is composed might cancel so that the star would
start from rest. It is interesting to note that by this
process the average initial velocity of a star composed of
A" constituents would vary ceteris paribus as JV~*, the
velocities of the constituents being assumed to be
haphazard. This might appear to lead to an equiparti-
tion of energy between stars of different masses, the mass
being proportional to N. But unless the number of
constituents is very small their velocities would have to be
enormous, and the suggestion does not seem to be tenable.
Moreover, it is difficult to see how a moving cluster could
be formed.*

An increase of velocity in. the next stage is the natural
re>ult of the central attraction of the stellar system. If
the star starts existence with no motion, it must start from
apcentron, and at all other points of the orbit its velocity
would be greater. After the star is old enough to have
described one quadrant of its orbit, we cannot look for
any increase in velocity from this cause ; in other words,
the progressive increase of velocity should cease after the
first 100,000,000 years. But we can scarcely compress
the development from Type B to Type M into that period.
Another difficulty is that the motion produced by the
central attraction would be mainly in the plane of the
galaxy ; there is no explanation of the motions perpendi-
cular to that plane acquired by the stars of the later types.
It does not seem possible to account for these extra-
galactic motions in any simple way.

I see no alternative to supposing that the K and M
stars have been formed originally in a more globular
distribution than the early type stars. It may be that the

* This suggestion (with the objections to it) was mentioned to me by
Prof. A. Schoster.

260 STELLAR MOVEMENTS CHAP.

birth of stars was for some reason retarded in the galactic
plane, and that this is the reason why early type stars
abound there. Perhaps a more likely supposition is that
massive stars with slow development have formed where
the material was rich, and small stars with rapid develop-
ment have formed where the star-stuff was scanty. Thus
the outlying parts of the stellar system away from the
galaxy have given birth to the small stars which have
rapidly reached the M stage ; and these, having fallen in
from a great distance, have acquired large velocities. The
regions of the galactic plane, richly supplied with the
necessary material, have formed large stars, which are
only slowly developing, and these have remained moving
in the galactic plane. The hypothesis in outline seems
fairly plausible ; but so long as the difficulty of the double
character of the Type M stars remains we cannot regard
any explanation as complete.

The foregoing suggestion is also applicable if we adopt
Russell's hypothesis (p. 170). His view is that only the
most massive stars are able to heat themselves up to the
high temperatures characteristic of Types B and A,
Accordingly, these types will only have originated
where the star-forming material was rich, and their
galactic concentration and small velocities can be
accounted for.

The problem of the equilibrium of the Milky Way is
another subject for reflection. It seems necessary to grant
that it is in some sort of equilibrium ; that is to say, the
individual stars do not oscillate to and fro across the stellar
system in a 300,000,000 year period, but remain concen-
trated in the clusters, which they now form. The only
possible explanation in terms of known forces is that the
Milky Way as a whole is in slow rotation, a condition
which has been considered by H. PoincareY 5 To obtain
some notion of the order of magnitude of the rotation, let
us assume the total mass of the inner stellar system to be

xii DYNAMICAL THEORY 261

10 9 times the Sun's mass, and the distance of the Milky
Way to be 2,000 parsecs ; the angular velocity for equili-
brium will then be 0"'5 per century. It may be pointed
out that Charlier 5 has found that the node of the invariable
plane of the solar system on the plane of the Milky Way
has a direct motion amounting to 0"'35 per century, a
motion which might equally well be expressed as a rotation
of the stars in the plane of the Milky Way in a retrograde
direction. Perhaps it would be straining the result
too far to regard this as evidencing the truth of our
surmise.

With this brief reference to the dynamical aspect of the
problem, we conclude our survey of the structure of the
stellar system. The results discussed have, with few
exceptions, come to light during the last fifteen years ;
but they are the outcome of a century's labour of
preparation. The proper motions now used are based on
observations which go back to the time of Bradley ; and
the modern instrumental methods, which are now yielding
parallaxes and proper motions for discussion, have a long
history of gradual development behind them. The
progress of stellar investigation must not be measured by
the few conclusions to which we have been able to give
definite statement. In the future the fruits of these
labours will be reached far more fully.

Meanwhile the knowledge that has been attained shows
only the more plainly how much there is to learn. The
perplexities of to-day foreshadow the discoveries of the
future. If we have still to leave the stellar universe a
region of hidden mystery, yet it seems as though, in our
exploration, we have been able to glimpse the outline of
some vast combination which unites even the farthest
stars into an organised system.

262 STELLAR MOVEMENTS CHAP, xn

REFERENCES. CHAPTER XII.

1. Cf. Poincare, 7/<//<//,;>r.s Cosmogcniques, p. '257.

2. Jeans, Monthly Notices, Vol. 74, p. 109.

3. Eddington, Monthly Notices, Vol. 74, p. 5.

4. Turner, Monthly Notices, Vol. 72, pp. 387, 474.

5. Charlier, Lund Meddelanden, Series 2, No. 9, p. 78.

6. Poincare, Hypotheses Cosmogoniques, p. 263.

INDEX

ABSORBING matter, 237

Absorption of light in space, 33, 203,

217

Adams, W. S., 63, 155
Age of moving clusters, 254
Aity's method (solar motion), 79
Aitken, K. G., 178
Antares, 45, 49
Apex, Solar

position of. 73, li>2

theory of determining, 79
Apices of star-streams 100, 123

summary of determinations, 125
Arcturus, 45, 51
Auwers, G. F. J. A., 207

B TYPE stars, 160, 179

Backhouse, T. W., 65

Barnard, E. E., 237

Beljawsky, S., 124

Bellamy, 'F A., 19

Bessel's method (solar motion), 83,

119

Betelgeuse, 49, 63
Bibliographies, 28, 70, 85, 153, 231
Binary stars, densities, 23, 174

evolution, 178

masses, 22, 160

number of, 44

relation between period and tvpe,
178

physical connection, 54
Bohi-; N., 8
Boss, B., 64
Boss, L., masses of binaries, 23

solar apex. 73, H>2

spectral types, 158, 167, 180

star-streams, 124

Taurus cluster, 56

Boss's 'Preliminary General Catalogue,'
18, 93

C CHAKACTKRISTK , 175

Campbell, W. W., distances of types,

167

Maxwellian law, 152
periods of binaries, 178

Campbell, W. W., solar motion, 73, 162

spectroscopic binaries, 44

star-streams, 107

velocities of Types, 156
Canopus, 45

a Centauri, 15, 19, 21, 22, 41, 50, 246
Cepheid variables, 6, 240
Chapman, S., 5, 186
Charlier, C. V. L., 124, 207, 261
Clusters, 241
Clusters, Moving, 54
Colour index, 11

Comstock, G. C., 19, 76, 124, 207
Convergence of stream motions, 99
Corona Austrina, 237
Counts of stars, 184
Cross proper motions, 158, 166
61 Cygni cluster, 70
Cvgnus region of Milky Way 235. 244,

258
C.Z. 5 h 24318, 20, 41, 50

DARK nebulas, 237
Densities of stars, 23, 174
Density of stellar distribution near the
Sun, 44

density law, 201, 214, 217
Distances of bright stars, 224

Carrington stars, 226

Magellanic Cloud, 240

Milky Way. 236

near stars/41, 47

spectral types, 167

two star-streams, 110, 139
Distribution of nebulae, 27, 241, 242
Distribution of stars in distance, 222

relative to Galactic plane, 165, 198
Donner, A. S., 60
Doppler's principle, 19
Double stars. See Binary stars
Draper classification, 7
Drift, definition of, 89
Drift O, 121
'Dwarf" stars, 170
Dynamics, stellar, 25! i
Dvson, F. W. , Carrington stars, 19,
225, 228

264

INDEX

Dyson, F. W., Groombridge stars, 75
'Stars of large proper motion, 42, 104,
124, 155

EASTON, C., 235, 244
Effective wave-lengths, 12
Ellipsoidal hypothesis, 103, 130

comparison with two-drift hypo-
thesis, 133, 229

radial velocities, 145
Encounters of stars, 249
Equilibrium of Milky Way, 260
Equipartition of energy, 159, 247
Error-law for luminosities, 173

parallaxes, 206

velocities, 127, 151

Errors of observation, effect of, 147, 229
Espin, T. E., 181
Evolution of binary systems, 178

spectral types, 7, 176

stellar system, 259

FATH, E. A., 27, 241
Fourier integrals, 209
Fourth type stars, 181
Fowler, A , 8
Franklin- Adams, J., 191
Frost, E. B., 44, 155
Functions, tables of /(T), 129

!), 132
141

GALACTIC longitude, variations of star

density, 196
Galactic plane, 31

concentration of stars, 191

concentration of special objects, 165,
239

relation of drifts to, 117
Galactic pole, position of, 239
Gas theory, 159, 247
"Giant" and "Dwarf" stars, 170
Gill, SirD., 181, 191
Gravitation, interstellar, 246
Groombridge 34 comes, 6
a Gruis, 181

HALM, J., constant of precession, 150
equipartition of energy, 159
two-drift theory, 117, 124
three-drift theory, 118, 121

Harvard Standard Sequence, 5

Helium stars, see B type stars

Herschel, Sir J., 192

Herschel, Sir W., 73, 238

Herschel's star-gauges, 191, 192, 213

prung, E.. '--characteristic, 175
effective wave-lengths, 12
"giant" and "dwarf" stars, 170
Lesser Magellanic Cloud, -Jin
objects with galactic concentration,

239
Ursa Major cluster. 61

Hinks, A. R., 240, 241
Hough, S. S., 117, 124, 150
Hyades, 56

INNES, R. T. A., 18, 156

Integral equations, 208, 219, 223
"Island universe" theory, 242

JKANS, J. H., 251
Jones, H. S., 167

KAPTEYN, J. C., distances of spectral
types, 167

fastest moving star, 18

luminosity and density-laws, 216

magnitude statistics, 191

parallaxes, 41,60

Perseus cluster. 64

speeds of stars,' 154, 156

star-streams, 87, 124, 164

Type N stars, 182
Kapteyn's mean parallaxes, 205
Kapteyn's Plan of Selected Areas, 26
King, E. S., 11
Kobold, H., 83, 119, 138
Kustner, F., 60

LANE-RITTER Theory, 173
Leavitt, Miss H. S. , 240
Lockyer, Sir J. N., 10, 174
Ludendorff, H., 10
Luminosities of stars, 45

dependence on spectral type, 48, 164

of Taurus cluster, 60
Luminosity law, 201

Dyson and Eddington's results, 227

Kapteyn's results, 216

Seeliger's results, 214

M TYPE stars, dual character of, 169
Magellanic Clouds, 239
Magnitudes of stars, 3

absolute magnitudes, set Luminosities.

stars of the two drifts, 109

statistics, 188
Masses of stars, 21, 160
Maury, Miss A. C., 10, 175.
Maxwellian law, 127, 151
Melotte, P. J., 5, 186
Milky Way, 196, 232

bright stars in, 234

distance of, 236

equilibrium of, 260

spiral theory, 244
Monck, W. H., i:>4
M t>t H.* /HI- it/ in rix, S(i
Moving clusters, 54

criteria for, 67

dissolution of, !'.").'{

gravitation in, 'J .">:;

iVi-rus duster, ")<i

Taurus cluster, 64

Ursa Major cluster, lil

INDEX

265

N TYPE stars, 181
Nebula?, dark, 237

distribution of. -~

irregular, -'\~

planetary, 15<>, '24 1

spiral, 242

Xi-wi-omb, S., 76, 234
Number of stars in the sky, 190, 195

OBLATENKSS of stellar system, 200
Observation, data of, "2
Observational errors, effect on drift-
constants, 147

formula for correcting statistic^ _'_". i
Orbits, stellar, time of describing, 255
Orion, 63, 238
Orion type of spectrum, 9, 159, 179

PAHLKN, E. v. D., 242
Parallactic motion, 79

for different magnitudes, 203, 207

for different types, 167
Parallaxes, 12 ; accuracy of, 15

Kapteyn's formulae, 205

large (tables), 41, 47

probable value, 222

stars of the two drifts, 113, 116

Taurus cluster, 60
Parkhurst, J. A., 181
Parsec, 14

Periods of binary stars, 178
Perseus cluster, 56
Photographic magnitudes, 3
Photometric magnitudes, 3
Pickering, E. C., 8,165, 194
Plan of Selected Areas, 26
Planetary nebulae, 156, 241
Pleiades, 63
Poincare, H. , 260
Precession, constant of, 73, 150
Preferential motions, 87
Proctor, R. A., 56
Proper motions, accuracy of, 18

analysis of, 88, 127

cross motions, 158, 166

large (list of), 19

mean proper motion curves, 112, 116,
139

solar motion from, 73, 79

statistical investigations of, 218

RADIAL motions in stellar system, 257
Radial Velocities, accuracy of, 20

comparison with error-law, 151

confirmation of star-streaming, 107

ellipsoidal hypothesis, 145

largest motions, 20

progression with spectral type, 156

solar motion from, 73, 84

two-drift hypothesis, 143
Regulus, 181
Rigel, 45
Rotation of 'Milky Way, 261

Rudolph, K., 124

Russell, H. X., 24, 17<>, 2m

SAGITTARIUS region of Milky Wav, 38,

197, 232, 258
Schuster, A., 259

Schwarzschikl, K., colour index, II
effective wave length, 12
ellipsoidal theory, 103, 124, 130
luminosity law, 217
magnitude statistics, 206
parallaxes of spectral types, 167
solution of statistical equations. _'( is
Scorpius-Centaurus cluster, 70
Scares, F. H., 5

Secchi's classification of spectra, 10
Seeliger, H., 213
Selected Areas, plan of, 26
Selection by magnitude, effect of, 46,

172,

Shapley, H., 24, 174
Sirius, 22, 41, 50, 61, 63
Sitter, W. de, 60

Skewness of velocity-distribution, 135
Solar motion, 71 ; dependence on spec-
tral type, 77, 162
direction of, 73, 102
theory of determining, 79
Spectral type, classification, 7
evolution, 176
in relation to density, 174
galactic plane, 165, 169, 259
luminosity, 48, 164
mass, 160
period, 178
solar motion, 77, 162
speed, 156, 259
star-drifts, 110, 163
Spectroscopic binaries, 20, 24, 44
Speed, see Velocity.
Spiral nebulae, 242
Star-ratio, 184
Star-streams, 86

mathematical theory, 127
suggested explanations, 244, 257
tabulated results, 124
Stellar system, dynamics of, 246
evolution of, 259
form and dimensions, 31
oblateness of, 200
Stumpe, O, 76
Sun, motion in space, see Solar

motion.

position in stellar system, 31, 196
stellar magnitude of, 6

TABLES,

(1) giving data for individual stars
largest proper motions, 19
masses (well-determined), 22
motions, transverse and radial, 50
parallaxes and luminosities, 41 , 47

266

INDEX

Tables, Perseus cluster, 66

stars with c-characteristic, 175 .
Ursa Major cluster, rt3

(2) <fii in;/ tteUisticnl rtxuff*
colour index and >pectral type, 11
densities of stars, 174

distance distribution of stars, 216,

221. 24, 226
galactic distribution, 165, 181,

239

luminosities and spectra, 48
luminosity-law, '2-27
mean parallaxes, Kapteyn's, 20.")

for spectra, 167

velocities, for spectra, 156, 158
number of stars in sky, 190

(log B m ) 189, 190, 213
periods of binaries, 178
radial motions and error-law, 151,

152
solar motion, for magnitude, 76

for spectra, 77, 162
star-streams, magnitudes, 109

mean parallaxes, 113

number of stars, 117

radial motions, 107

spectra, 110

(3) il/tiNtraf.iit.<i method* and theory
comparison of hypotheses, 134
star-stream analysis, 91, 116, 133,

143

(4) mathematical fmn-tions
/(T), 129
/)//(-. 132

ff (T), 141
(.")) .iii)imnry of ^tar-stream dfs-

'>#

Api<-es of drifts, 125
Vertex, 121

Taurus clust.-r. 5ii, 253
Thackeray, \V. <;.. 7-~>
Three-drift hypothesis. 12i
Turner, H. H., magnitude statistics,

193
parsec, 14

Turner, H. H., proper motions, 19
theory of star-streaming, 257
Ursa Major system, 62

Two-drift hypothesis, 90, 127

comparison with ellipsoidal hypo-
theses, 133, 22! l
radial velocities, 143

Two drifts, diagrams, 95
distribution in the sky, 117
division of stars between, 117
in nearest stars, 53
in stars of large proper motion, 1(>5
magnitudes of stars in, 109
mathematical theory, 127
mean distances of, 111
radial motions, 106
spectra of stars in, 110
suggested explanations, 244, 257
summary of determinations, 124

UNIT of distance, 14

of drift- velocity (1 /h), 128

of luminosity, 6
Ursa Major cluster, 61

VARIABLES, Cepheid, 6, 240
Variables, Eclipsing, 24, 174, 239
Velocities of stars, dependence on

distance, 161

dependence on type, 156, 259
gravitational effects on, 246
of intrinsically faint stars, 52
of nearest stars. .">u
origin of, 258

Velocity-ellipsoid, 137, 145
Velocity-law, 201, 220
Vertex of star-streaming, 102

determinations of, 124
Visual magnitudes, 3, 193

\V.\VK I.I:N<;THS, effective, 12
Weersma, H A!, 41, 145, 103
Whirlpool Nebula, 241
Wolf, M., 238
Wolf-Rayet stars, 8, 239

R. fLAY AND !< > > ' WW K ST.. MAMI"I'.I> ft., B.H , AM- I:' S'.AV

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